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Notes Introduction In this unit, students will be introduced to additional nonlinear functions such as radical and rational equations. Students learn how to simplify radical and rational expressions, and how to solve equations involving these expressions. Students also explore triangles through the Pythagorean Theorem and trigonometric ratios. Nonlinear functions such as radical and rational functions can be used to model real-world situations such as the speed of a roller coaster. In this unit, you will learn about radical and rational functions. Assessment Options Unit 4 Test Pages 779–780 of the Chapter 12 Resource Masters may be used as a test or review for Unit 4. This assessment contains both multiple-choice and short answer items. TestCheck and Worksheet Builder This CD-ROM can be used to create additional unit tests and review worksheets. 582 Unit 4 Radical and Rational Functions 582 Unit 4 Radical and Rational Functions Radical and Rational Functions Chapter 11 Radical Expressions and Triangles Chapter 12 Rational Expressions and Equations Teaching Suggestions Have students study the USA TODAY Snapshot. • Ask students how the data in the graph differs from that of a linear function. The data in the graph does not follow a straight line, therefore it is not linear. • Why might math be important to a roller coaster designer? Sample answer: A roller coaster designer might use math to calculate the speed of a roller coaster before it is built to make sure it is safe. Building the Best Roller Coaster USA TODAY Snapshots® Each year, amusement park owners compete to earn part of the billions of dollars Americans spend at amusement parks. Often the parks draw customers with new taller and faster roller coasters. In this project, you will explore how radical and rational functions are related to buying and building a new roller coaster. Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 4. A day in the park What the typical family of four pays to visit a park1: $163 $180 11-1 590 12-2 652 $116 $120 $60 0 Lesson Page Additional USA TODAY Snapshots appearing in Unit 4: Chapter 11 Global spending on construction (p. 615) Chapter 12 Most Americans have one or two credit cards (p. 672) Cost of parenthood rising (p. 689) ’93 ’95 ’97 ’99 ’01 1 — Admission for two adults and two children, parking for one car and purchase of two hot dogs, two hamburgers, four orders of fries, four small soft drinks and two children’s T-shirts. Source: Amusement Business Unit 4 By Marcy E. Mullins, USA TODAY Radical and Rational Functions 583 Internet Project A WebQuest is an online project in which students do research on the Internet, gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance to the next step in their WebQuest. At the end of Chapter 12, the project culminates with a presentation of their findings. Teaching notes and sample answers are available in the WebQuest and Project Resources. Unit 4 Radical and Rational Functions 583 Radical Expressions and Triangles Chapter Overview and Pacing PACING (days) Regular Block LESSON OBJECTIVES Basic/ Average Advanced Basic/ Average Advanced Simplifying Radical Expressions (pp. 586–592) • Simplify radical expressions using the Product Property of Square Roots. • Simplify radical expressions using the Quotient Property of Square Roots. 2 2 1.5 1.5 Operations with Radical Expressions (pp. 593–597) • Add and subtract radical expressions. • Multiply radical expressions. 2 2 1 1 Radical Equations (pp. 598–604) • Solve radical equations. • Solve radical equations with extraneous solutions. Follow-Up: Use a graphing calculator to graph radical equations. 2 3 (with 11-3 Follow-Up) 1 2 The Pythagorean Theorem (pp. 605–610) • Solve problems by using the Pythagorean Theorem. • Determine whether a triangle is a right triangle. 1 1 0.5 0.5 The Distance Formula (pp. 611–615) • Find the distance between two points on the coordinate plane. • Find a point that is a given distance from a second point in a plane. 1 1 0.5 0.5 Similar Triangles (pp. 616–621) • Determine whether two triangles are similar. • Find the unknown measures of sides of two similar triangles. 1 1 0.5 0.5 Trigonometric Ratios (pp. 622–630) Preview: Use paper triangles to investigate trigonometric ratios. • Define the sine, cosine, and tangent ratios. • Use trigonometric ratios to solve right triangles. 2 2 1 1 Study Guide and Practice Test (pp. 632–637) Standardized Test Practice (pp. 638–639) 1 1 0.5 0.5 Chapter Assessment 1 1 0.5 0.5 13 14 7 8 TOTAL Pacing suggestions for the entire year can be found on pages T20–T21. 584A Chapter 11 Radical Expressions and Triangles Timesaving Tools ™ All-In-One Planner and Resource Center Chapter Resource Manager 643–644 645–646 647 648 649–650 651–652 653 654 655–656 657–658 659 660 661–662 663–664 665 666 667–668 669–670 671 672 673–674 675–676 677 678 700 679–680 681–682 683 684 Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ) Ass ess me nt Pre req u Wo isite rkb Ski ook lls Enr ich me nt S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice Ave rag e) Rea di Ma ng to the ma Learn tics CHAPTER 11 RESOURCE MASTERS See pages T12–T13. SC 21 83 11-1 11-1 84 11-2 11-2 GCS 43 85 11-3 11-3 GCS 44, SC 22 86 11-4 11-4 SM 85–90 87 11-5 11-5 88 11-6 11-6 700 89 11-7 11-7 685–698, 702–704 90 699 37–38 699, 701 61–62 Materials graphing calculator graphing calculator (Follow-Up: graphing calculator) 32 (Follow-Up: ruler, grid paper, protractor), string, drinking straw, paper clip, protractor, tape, meter sticks or tape measure *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 11 Radical Expressions and Triangles 584B Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 2, students learned to find principal square roots. They solved proportions in Chapter 3. Students found the factors of the difference of squares in Chapter 8. This Chapter In this chapter, students simplify and perform operations with radical expressions. They apply these skills to solving radical equations. The Pythagorean Theorem is used to solve problems involving right triangles. The Distance Formula is introduced as an application of the Pythagorean Theorem. Similar triangles and trigonometric ratios are used to find missing measures in triangles. Future Connections The basics of trigonometry and the study of triangles are explored in this chapter. These basics will be applied to deeper studies of triangles in later mathematics courses. Many of these concepts are used in construction and engineering. 584C Chapter 11 Radical Expressions and Triangles Simplifying Radical Expressions A radical expression contains a square root. The expression is in simplest form if the expression inside the radical sign, or radicand, has only 1 as a perfect square factor. The Product Property of Square Roots states that the square root of a product equals the product of each square root. Prime factorizations combined with the Product Property of Square Roots can be used to simplify radical expressions. Principal square roots are never negative, so absolute value symbols must be used to signify that some results are not negative. The Quotient Property of Square Roots states that the square root of a quotient equals the quotient of each square root. The Quotient Property of Square Roots can be used to derive the Quadratic Formula. A fraction does not have a radical in its denominator if it is in simplest form. Since squaring and taking a square root are inverse functions, you multiply the numerator and the denominator by the same number so that the denominator contains a perfect square. Remember that the numerator must be multiplied by the same amount so that the whole fraction is being multiplied by a value of 1. This process is called rationalizing the denominator. After rationalizing the denominator, check for coefficients of the radical in the numerator that will simplify with the denominator. If the denominator is an expression containing a radical, multiply by the conjugate. For example, if the denominator is a b, multiply by a b. Operations with Radical Expressions Use the process of combining like terms to simplify expressions in which radicals are added or subtracted. For terms to be combined, their radicands must be the same. As in combining monomials with variables, only the coefficients of the radicals are combined. Be sure to simplify all radicals first. When multiplying radical expressions, first multiply the coefficients, and then use the Product Property of Square Roots to multiply the radicals. Simplify each term and then combine like terms as necessary. Radical Equations Equations that contain variables in the radicand are called radical equations. To solve radical equations, the radical must first be isolated on one side. Then square each side. This will eliminate the radical. This process sometimes produces results that are not solutions of the original equation. These are called extraneous solutions. All solutions must be substituted back into the original equation to check their validity. The Pythagorean Theorem The two sides of a right triangle that form the right angle are the legs of the triangle. The third, and longest, side is the hypotenuse. The hypotenuse is the longest side because it is always across from the angle with the greatest measure. The Pythagorean Theorem states that the sum of the squares of the legs equals the square of the hypotenuse. The formula is c2 a2 b2, where a and b are the measures of the legs and c is the measure of the hypotenuse. This formula can be used to find the length of any missing side of a right triangle if the lengths of the other two sides are known. Any three whole numbers that satisfy this equation are known as a Pythagorean Triple. These triples represent side lengths that always form right triangles. It follows that if three numbers do not satisfy the Pythagorean Theorem, then sides of their length will not form a right triangle. Trigonometric Ratios Trigonometry is the mathematical study of angles and triangles. Ratios comparing the measures of two sides of a right triangle are called trigonometric ratios. The three most common trigonometric ratios are sine, cosine, and tangent. These ratios can be used to find the measures of missing sides or the measures of the acute angles. The relationship of the two sides necessary for the problem to a specific acute angle determines which ratio is used. If the measures of just two sides of a triangle or the measures of one side and one acute angle are known, then the measures of all of the rest of the sides and angles can be found. This is called solving the triangle. Trigonometric ratios are used to find distances in problems involving angles of elevation and angles of depression. The Distance Formula If the Pythagorean Theorem is solved for c, the result is the Distance Formula. The variable a is expressed as the difference of the x-coordinates of the endpoints of the hypotenuse and b is expressed as the difference of the y-coordinates. This formula is used to find the distance between any two points. You can also find one missing coordinate of an endpoint if you know the other coordinate, the coordinates of the other endpoint, and the distance between the two points. Similar Triangles Similar triangles have the same shape, but are not necessarily the same size. All of the corresponding angles will have equal measures, and the corresponding sides will all be proportional. If the sides have a 1 to 1 ratio of proportionality, then the similar triangles are the same size. When determining whether triangles are similar, all you need to check is if the corresponding angles have the same measure. If all the angle measures cannot be determined, then check the corresponding sides to see if they are proportional. Proportions can be used to find the lengths of missing sides of similar triangles. You must know the lengths of at least one pair of corresponding sides and the length of the side that corresponds to the missing side’s length. Set up the proportion, and then cross multiply. Solve the resulting equation. Chapter 11 Radical Expressions and Triangles 584D and Assessment ASSESSMENT INTERVENTION Type Student Edition Teacher Resources Ongoing Prerequisite Skills, pp. 585, 592, 597, 603, 610, 615, 621 Practice Quiz 1, p. 603 Practice Quiz 2, p. 621 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 37–38, 61–62 Quizzes, CRM pp. 699–700 Mid-Chapter Test, CRM p. 701 Study Guide and Intervention, CRM pp. 643–644, 649–650, 655–656, 661–662, 667–668, 673–674, 679–680 Mixed Review pp. 592, 597, 603, 610, 615, 621, 630 Cumulative Review, CRM p. 702 Error Analysis Find the Error, pp. 600, 618 Find the Error, TWE pp. 600, 618 Unlocking Misconceptions, TWE p. 612 Tips for New Teachers, TWE pp. 587, 624 Standardized Test Practice pp. 591, 597, 602, 606, 608, 610, 615, 620, 630, 637, 638–639 TWE pp. 638–639 Standardized Test Practice, CRM pp. 703–704 Open-Ended Assessment Writing in Math, pp. 591, 597, 602, 610, 614, 620, 630 Open Ended, pp. 589, 595, 600, 607, 612, 618, 627 Standardized Test, p. 639 Modeling: TWE pp. 603, 610, 621 Speaking: TWE pp. 597, 615 Writing: TWE pp. 592, 630 Open-Ended Assessment, CRM p. 697 Chapter Assessment Study Guide, pp. 632–636 Practice Test, p. 637 Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 685–690 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 691–696 Vocabulary Test/Review, CRM p. 698 Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. 584E Chapter 11 Radical Expressions and Triangles Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson 11-4 AlgePASS Lesson 32 Solving Problems Using the Pythagorean Theorem ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 585 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 589, 595, 600, 607, 612, 618, 627) • Reading Mathematics, p. 631 • Writing in Math questions in every lesson, pp. 591, 597, 602, 610, 614, 620, 630 • Reading Study Tip, pp. 586, 611, 616, 623 • WebQuest, p. 590 Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 83–90 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test For more information on Intervention and Assessment, see pp. T8–T11. Teacher Wraparound Edition • Foldables Study Organizer, pp. 585, 632 • Study Notebook suggestions, pp. 589, 595, 600, 608, 613, 618, 627, 631 • Modeling activities, pp. 603, 610, 621 • Speaking activities, pp. 597, 615 • Writing activities, pp. 592, 630 • Differentiated Instruction, (Verbal/Linguistic), p. 599 • ELL Resources, pp. 584, 591, 596, 599, 602, 609, 614, 620, 629, 631, 632 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 11 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 11 Resource Masters, pp. 647, 653, 659, 665, 671, 677, 683) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 3.2, 7.9, 7.10, 8.6 For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 11 Radical Expressions and Triangles 584F Radical Expressions and Triangles Notes Have students read over the list of objectives and make a list of any words with which they are not familiar. • Lessons 11-1 and 11-2 Simplify and perform operations with radical expressions. • Lesson 11-3 Solve radical equations. • Lessons 11-4 and 11-5 Use the Pythagorean Theorem and Distance Formula. • Lessons 11-6 and 11-7 Use similar triangles and trigonometric ratios. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson. Key Vocabulary • • • • • radical expression (p. 586) radical equation (p. 598) Pythagorean Theorem (p. 605) Distance Formula (p. 611) trigonometric ratios (p. 623) Physics problems are among the many applications of radical equations. Formulas that contain the value for the acceleration due to gravity, such as free-fall times, escape velocities, and the speeds of roller coasters, can all be written as radical equations. You will learn how to calculate the time it takes for a skydiver to fall a given distance in Lesson 11-3. Lesson 11-1 11-2 11-3 11-3 Follow-Up 11-4 11-5 11-6 11-7 Preview 11-7 NCTM Standards Local Objectives 1, 2, 6, 8, 9, 10 1, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8 1, 2, 3, 6, 8, 9, 10 1, 2, 3, 6, 8, 9, 10 1, 2, 3, 6, 8, 9, 10 1, 3, 7 3, 6, 8, 9, 10 Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 584 584 Chapter 11 Radical Expressions and Triangles Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 11 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 11 test. Chapter 11 Radical Expressions and Triangles Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 11. For Lessons 11-1 and 11-4 This section provides a review of the basic concepts needed before beginning Chapter 11. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 37–38 and 61–62. Find Square Roots Find each square root. If necessary, round to the nearest hundredth. (For review, see Lesson 2-7.) 1. 25 5 2. 80 8.94 3. 56 7.48 4. 324 18 For Lesson 11-2 Combine Like Terms Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson. Simplify each expression. (For review, see Lesson 1-6.) 5. 3a 7b 2a a 7b 6. 14x 6y 2y 14x 4y 7. (10c 5d) (6c 5d) 16c 8. (21m 15n) (9n 4m) 25m 6n For Lesson 11-3 Solve Quadratic Equations Solve each equation. (For review, see Lesson 9-3.) 10. x2 10x 24 0 {6, 4} 1 12. 2x2 x 1 2 , 1 2 9. x(x 5) 0 {0, 5} 11. x2 6x 27 0 {3, 9} For Lesson 11-6 Proportions Use cross products to determine whether each pair of ratios forms a proportion. Write yes or no. (For review, see Lesson 3-6.) 2 8 13. , 3 12 yes 4 16 14. , 5 25 no 8 12 15. , 10 16 6 3 16. , 30 15 no For Lesson Prerequisite Skill 11-2 11-3 Multiplying Binomials (p. 592) Finding Special Products (p. 597) Evaluating Radical Expressions (p. 603) Simplifying Radical Expressions (p. 610) Solving Proportions (p. 615) Evaluating Expressions (p. 621) 11-4 yes 11-5 11-6 11-7 Make this Foldable to help you organize information about radical expressions and equations. Begin with a sheet of 1 plain 8" by 11" paper. 2 Fold in Half Fold Again Fold the top to the bottom. Fold in half lengthwise. Cut Label Open. Cut along the second fold to make two tabs. Label each tab as shown. Radical Expression s Radical Equations As you read and study the chapter, write notes and examples for each lesson under each tab. Reading and Writing Chapter 11 Radical Expressions and Triangles 585 TM For more information about Foldables, see Teaching Mathematics with Foldables. Organization of Data: Annotating As students read and work through the chapter, have them make annotations under the tabs of their Foldable. Explain to them that annotations are usually notes taken in the margins of books we own to organize the text for review or studying. Annotations often include questions that arise as we read the chapter, reader comments and reactions, sentence length summaries, steps or data numbered by the reader, and key points highlighted or underlined. Chapter 11 Radical Expressions and Triangles 585 Simplifying Radical Expressions Lesson Notes 1 Focus 5-Minute Check Transparency 11-1 Use as a quiz or review of Chapter 10. Mathematical Background notes are available for this lesson on p. 584C. • Simplify radical expressions using the Product Property of Square Roots. • Simplify radical expressions using the Quotient Property of Square Roots. • radical expression • radicand • rationalizing the denominator • conjugate A spacecraft leaving Earth must have a velocity of at least 11.2 kilometers per second (25,000 miles per hour) to enter into orbit. This velocity is called the escape velocity. The escape velocity of an object is given by the radical expression 2GM , where G is the gravitational constant, R M is the mass of the planet or star, and R is the radius of the planet or star. Once values are substituted for the variables, the formula can be simplified. Building on Prior Knowledge Students were introduced to the Quadratic Formula in Lesson 10-4 and they learned how to use it to solve quadratic equations. In this lesson, students learn how to derive the Quadratic Formula from the standard form of a quadratic equation using the Quotient Property of Square Roots. are radical expressions used in space exploration? Ask students: • In the formula for escape velocity, what does the radical sign mean? The radical sign means that you must find the square root of the value under the radical sign. • Based on what you know about the order of operations, how do you think you should simplify the radical expression in the escape velocity formula? You should simplify the expression under the radical sign before finding the square root. are radical expressions used in space exploration? Vocabulary PRODUCT PROPERTY OF SQUARE ROOTS A radical expression is an expression that contains a square root. A radicand, the expression under the radical sign, is in simplest form if it contains no perfect square factors other than 1. The following property can be used to simplify square roots. Product Property of Square Roots • Words For any numbers a and b, where a 0 and b 0, the square root of a product is equal to the product of each square root. • Symbols ab a b • Example 4 25 4 25 The Product Property of Square Roots and prime factorization can be used to simplify radical expressions in which the radicand is not a perfect square. Example 1 Simplify Square Roots Simplify. a. 12 Study Tip Reading Math 23 is read two times the square root of 3 or two radical three. 12 2 2 3 22 3 23 Prime factorization of 12 Product Property of Square Roots Simplify. b. 90 90 23 35 32 25 310 Prime factorization of 90 Product Property of Square Roots Simplify. 586 Chapter 11 Radical Expressions and Triangles Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 643–644 • Skills Practice, p. 645 • Practice, p. 646 • Reading to Learn Mathematics, p. 647 • Enrichment, p. 648 Parent and Student Study Guide Workbook, p. 83 School-to-Career Masters, p. 21 Transparencies 5-Minute Check Transparency 11-1 Answer Key Transparencies Technology Interactive Chalkboard The Product Property can also be used to multiply square roots. Study Tip Alternative Method To find 3 15 , you could multiply first and then use the prime factorization. 45 3 15 32 5 35 2 Teach Example 2 Multiply Square Roots Find 3 15 . 3 3 5 3 15 32 5 35 PRODUCT PROPERTY OF SQUARE ROOTS Product Property of Square Roots Product Property Simplify. Intervention In order to simplify square roots with the Product Property of Square Roots, students need to be able to find the prime factorization of the radicand. Take a few minutes to review finding prime factorizations. After a quick refresher, students can focus on learning the new concept, rather than trying to recall earlier material. New When finding the principal square root of an expression containing variables, be x2. It may seem that sure that the result is not negative. Consider the expression x2 x. Let’s look at x 2. x2 x 2 2 (2) 4 2 2 2 Replace x with 2. (2)2 4 4 2 For radical expressions where the exponent of the variable inside the radical is even and the resulting simplified exponent is odd, you must use absolute value to ensure nonnegative results. x2 x x3 xx x4 x2 x5 x2x x6 x3 Example 3 Simplify a Square Root with Variables Simplify 40x4y5 z3. 40x4y5 z3 23 5 x4 y5 z3 In-Class Examples Prime factorization 22 2 5 x4 y4 y z2 z Product Property 1 Simplify. 2 2 5 x2 y2 y z z a. 52 213 Simplify. The absolute value of z ensures a nonnegative result. 2x2y2z10yz Power Point® b. 72 62 Teaching Tip Remind students that since 3 is a prime number, it cannot be factored further using prime factorization. QUOTIENT PROPERTY OF SQUARE ROOTS You can divide square roots and simplify radical expressions that involve division by using the Quotient Property of Square Roots. 2 Find 2 24 . Quotient Property of Square Roots • Words For any numbers a and b, where a 0 and b 0, the square root of a quotient is equal to the quotient of each square root. • Symbols a ba b • Example 43 3 Simplify 45a4b5 c6. 3a 2b 2 |c 3 | 5b 49 49 4 4 Study Tip Look Back To review the Quadratic Formula, see Lesson 10-4. You can use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the quadratic equation ax2 bx c 0. Interactive ax2 bx c 0 Original equation b a Chalkboard c a x2 x 0 Divide each side by a, a 0. (continued on the next page) www.algebra1.com/extra_examples PowerPoint® Presentations Lesson 11-1 Simplifying Radical Expressions 587 This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools Lesson 11-1 Simplifying Radical Expressions 587 In-Class Example 11y 33y b. 9 27 3 6 8 c. 4 c Subtract from each side. b2 b a a b2 c a 4a 2 x 2ba 2 2a 4a 4ac b2 4a b a 2 b2 4ac 2 Remove the absolute value symbols and insert . 2 Plus or Minus Symbol b 2a 2 4ac b 4a x 2 The symbol is used with the radical expression since both square roots lead to solutions. Quotient Property of Square Roots 2 4ac b x b 2a b2 4a Take the square root of each side. 2 Study Tip 4a Factor x2 x 2. x 2a 4a b b 4ac x 2a 4a b 2 b b Complete the square; 2. x2 x 2 2 Power Point® 4 Simplify. 12 215 a. 5 5 c a b a x2 x QUOTIENT PROPERTY OF SQUARE ROOTS 4a2 2a 2a b b 4ac x 2 2a b Subtract from each side. 2a Thus, we have derived the Quadratic Formula. A fraction containing radicals is in simplest form if no prime factors appear under the radical sign with an exponent greater than 1 and if no radicals are left in the denominator. Rationalizing the denominator of a radical expression is a method used to eliminate radicals from the denominator of a fraction. Example 4 Rationalizing the Denominator Simplify. 7x b. 10 a. 8 3 10 3 10 3 3 3 3 . Multiply by 30 Product Property of Square Roots 3 3 7x 7x 2 2 8 2 2 7x 2 . Multiply by 14x Product Property of Square Roots 22 4 2 c. Prime factorization 2 2 6 2 6 2 6 6 6 12 6 Product Property of Square Roots 2 2 3 6 Prime factorization 23 22 2 3 Divide the numerator and denominator by 2. 6 6 3 588 6 . Multiply by Chapter 11 Radical Expressions and Triangles Differentiated Instruction Logical Have students use the same method that is shown for the derivation of the Quadratic Formula to solve actual quadratic equations. Have students turn back to Lesson 10-4 and solve an example problem using this method. 588 Chapter 11 Radical Expressions and Triangles Binomials of the form pq rs and pq rs are called conjugates. For example, 3 2 and 3 2 are conjugates. Conjugates are useful when simplifying radical expressions because if p, q, r, and s are rational numbers, their product is always a rational number with no radicals. Use the pattern (a b)(a b) a2 b2 to find their product. 3 2 3 2 32 2 2 9 2 or 7 In-Class Example Power Point® 3 5 Simplify . 15 32 23 5 2 a 3, b 2 3 Practice/Apply 2 2 2 2 or 2 Example 5 Use Conjugates to Rationalize a Denominator 2 Simplify . 6 3 6 3 2 2 6 3 6 3 6 3 26 3 2 62 3 (a b)(a b) 12 23 36 3 3 2 3 12 23 Simplify. 33 Study Notebook 6 3 1 6 3 a2 Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 11. • include examples of how to use the Product and Quotient Properties of Square Roots to simplify radical expressions. • include examples of how to rationalize denominators, with and without conjugates. • include any other item(s) that they find helpful in mastering the skills in this lesson. b2 When simplifying radical expressions, check the following conditions to determine if the expression is in simplest form. Simplest Radical Form A radical expression is in simplest form when the following three conditions have been met. 1. No radicands have perfect square factors other than 1. 2. No radicands contain fractions. 3. No radicals appear in the denominator of a fraction. Answer Concept Check Guided Practice GUIDED PRACTICE KEY Exercises Examples 4 5, 6, 13 7, 8 9, 10, 14 11, 12 1 2 3 4 5 1. Both x 4 and x 2 are positive even if x is a negative number. x4 x2. See margin. 1. Explain why absolute value is not necessary for 1 1 a for a 0. a a 2. Show that a a a a a 3. OPEN ENDED Give an example of a binomial in the form ab cd and its conjugate. Then find their product. Sample answer: 22 33 and Simplify. 8. 2x2y315x 22 33; 19 4. 20 25 5. 2 8 4 7. 54a2b2 3ab6 8. 60x5y6 10. 3 10 30 10 6. 310 410 120 8 83 2 11. 7 3 2 4 6 2 9. 6 3 25 10 25 12. 2 4 8 Lesson 11-1 Simplifying Radical Expressions 589 Lesson 11-1 Simplifying Radical Expressions 589 Applications About the Exercises … Organization by Objective • Product Property of Square Roots: 15–26, 39, 41 • Quotient Property of Square Roots: 27–38, 40 Odd/Even Assignments Exercises 15–40 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 15–41 odd, 42, 43, 50–53, 62–89 Average: 15–41 odd, 45–47, 50–53, 62–89 (optional: 54–61) Advanced: 16–40 even, 44–83 (optional: 84–89) 14. PHYSICS The period of a pendulum is the time required for it to make one complete swing back and forth. The formula of the period P of a pendulum 32 ᐉ is P 2 , where ᐉ is the length of the pendulum in feet. If a pendulum in a clock tower is 8 feet long, find the period. Use 3.14 for . 3.14 s ★ indicates increased difficulty Practice and Apply Homework Help 51. A lot of formulas and calculations that are used in space exploration contain radical expressions. Answers should include the following. • To determine the escape velocity of a planet, you would need to know its mass and the radius. It would be very important to know the escape velocity of a planet before you landed on it so you would know if you had enough fuel and velocity to launch from it to get back into space. • The astronomical body with the smaller radius would have a greater escape velocity. As the radius decreases, the escape velocity increases. Simplify. 25. 7x3y33y 26. 6xy2z22xz For Exercises See Examples 15. 18 32 16. 24 26 15–18, 41, 44–46 19–22, 39, 40, 48, 49 23–26 27–32, 42, 43, 47 33–38 1 18. 75 53 19. 5 6 30 20. 3 8 26 2 21. 730 26 845 22. 23 527 90 23. 40a4 2a210 3 4 24. 50m3n5 5mn 22mn 147x6y7 25. 5 27. Extra Practice See page 844. x3xy 32. 3 Answer 13. GEOMETRY A square has sides each measuring 27 feet. Determine the area of the square. 28 ft2 2y 5 10 2 34. 2 35. 27 22 26 23 36. 3 The speed of a roller coaster can be determined by evaluating a radical expression. Visit www.algebra1.com/ webquest to continue work on your WebQuest project. 16 123 37. 11 521 335 38. 10 6 27 73 3 27 33 30. p p 28. 31. 2 26. 72x3y4 z5 310 35 64 10 5c c 5cd 4d 2d 5 2 5 3 18 54 92 33. 17 6 2 25 34. 2 36. 3 6 4 37. 4 3 3 17. 80 45 4 8 29. 8t 2t 4 5y 9x 12x y 32. 2 6 10 35. 7 2 37 ★ 38. 53 35 39. GEOMETRY A rectangle has width 35 centimeters and length 4 10 centimeters. Find the area of the rectangle. 602 or 84.9 cm2 a a 40. GEOMETRY A rectangle has length meters and width meters. 8 2 What is the area of the rectangle? a 2 m 4 41. GEOMETRY The formula for the area A of a square with side length s is A s2. Solve this equation for s, and find the side length of a square having an area of 72 square inches. s A ; 62 in. PHYSICS For Exercises 42 and 43, use the following information. 1 2 The formula for the kinetic energy of a moving object is E mv2, where E is the kinetic energy in joules, m is the mass in kilograms, and v is the velocity in meters per second. 2E 42. Solve the equation for v. v m 43. Find the velocity of an object whose mass is 0.6 kilogram and whose kinetic energy is 54 joules. 6 5 or 13.4 m/s 44. SPACE EXPLORATION Refer to the application at the beginning of the lesson. Find the escape velocity for the Moon in kilometers per second if 6.7 1020 km s kg 22 kg, and R 1.7 103 km. How does this G 2 , M 7.4 10 compare to the escape velocity for Earth? 2.4 km/s; The Moon has a much lower 590 escape velocity than Earth. Chapter 11 Radical Expressions and Triangles Teacher to Teacher Judy Buchholtz Dublin Scioto H.S., Dublin, OH “To make algebra more meaningful to my students, I bring in the school police officer to discuss the formula used in Exercises 45–47.” 590 Chapter 11 Radical Expressions and Triangles INVESTIGATION For Exercises 45–47, use the following information. NAME ______________________________________________ DATE 2d 45. Write a simplified expression for the speed if f 0.6 for a wet asphalt road. 3 46. What is a simplified expression for the speed if f 0.8 for a dry asphalt road? Example 1 Simplify 180 . Prime factorization of 180 Product Property of Square Roots Simplify. Simplify. Simplify 120a2 b5 c4. 120a2 b5 c4 3 3 2 b5 2 5 a c4 2 2 2 3 5 a2 b4 b c4 2 2 3 5 | a | b2 b c2 2 | a | b2c230b GEOMETRY For Exercises 48 and 49, use the following information. Hero’s Formula can be used to calculate the area A of a triangle given the three side lengths a, b, and c. Exercises Simplify. 1. 28 1 2 2. 68 27 s(s a )(s b )(s c), where s (a b c) A 5. 162 48. Find the value of s if the side lengths of a triangle are 13, 10, and 7 feet. 15 49. Determine the area of the triangle. 203 or 34.6 ft2 50. CRITICAL THINKING For any numbers a and b, where a 0 and b 0, ab a b . Product Property of Square Roots Example 2 51.4 mph 46. 26d Insurance investigators decide whether claims are covered by the customer’s policy, assess the amount of loss, and investigate the circumstances of a claim. The Product Property of Square Roots and prime factorization can be used to simplify expressions involving irrational square roots. When you simplify radical expressions with variables, use absolute value to ensure nonnegative results. 180 2 2 33 5 22 32 5 2 3 5 65 47. An officer measures skid marks that are 110 feet long. Determine the speed of the car for both wet road conditions and for dry road conditions. 44.5 mph, Insurance Investigator p. 643 (shown) and p. 644 Simplifying Radical Expressions Product Property of Square Roots Lesson 11-1 Police officers can use the formula s 30fd to determine the speed s that a car was traveling in miles per hour by measuring the distance d in feet of its skid marks. In this formula, f is the coefficient of friction for the type and condition of the road. ____________ PERIOD _____ Study Guide andIntervention Intervention, 11-1 Study Guide and 6. 3 6 7. 2 5 92 32 10. 9x4 2 |a | 4. 75 215 9. 4a2 10 52 12. 128c6 8 |c 3 | 2 10a 23 3x 2 15. 20a2b4 9 |x3| 2415 53 8. 5 10 11. 300a4 14. 3x2 3 3x4 13. 410 36 a 1 a 1 Simplify . a2 3a 1 a 1 a 3. 60 217 16. 100x3y 2 |a |b 25 10 | x |xy 17. 24a4b2 18. 81x4y2 19. 150a2 b2c 3c2 20. 72a6b 5z8 21. 45x2y 6z2 22. 98x4y 2a 2 | b | 6 9x 2 | y | 6 |a 3bc | 2b 5 |ab | 6c 3 |x |y 2z 45y 7x 2 |y 3z | 2 NAME ______________________________________________ DATE Online Research For more information about a career as an insurance investigator, visit: www.algebra1.com/ careers Source: U.S. Department of Labor 51. WRITING IN MATH Skills Practice, 11-1 Practice (Average) Simplify. How are radical expressions used in space exploration? Include the following in your answer: • an explanation of how you could determine the escape velocity for a planet and why you would need this information before you landed on the planet, and • a comparison of the escape velocity for two astronomical bodies with the same mass, but different radii. 1. 24 26 2. 60 215 3. 108 63 4. 8 6 43 5. 7 14 72 6. 312 56 902 32a3 B 48a3 C 64a3 D 96a3 10. 108x6 y4z5 6| x 3 | y 2z 2 3z 9. 50p5 5p 22p 23 12. 8 6 4o5 2| m | n 2o 214o 11. 56m2n 15 15. 5 13. 19. 1 7 4 4y 3y2 2 9ab 4ab4 15 32 23 3 21. 2 12 43 3 8 22. 3 3 53 57 4 921 37 37 1 27 26 5 23. 7 3 7 11 18 x3 23y 3|y| 5 2 5 32 14. 4 5 3k 8 3 10 8 11 16. 11 32x 18. x 3b 20. 2b 2 5 10 5 6k 17. 52. If the cube has a surface area of 96a2, what is its volume? C A 8. 27su3 3| u | 3su 7. 43 318 366 3 4 Standardized Test Practice ____________ PERIOD _____ p. 645 and Practice, 646Expressions (shown) Simplifyingp. Radical Answer the question that was posed at the beginning of the lesson. See margin. 24. 25. SKY DIVING When a skydiver jumps from an airplane, the time t it takes to free fall a given distance can be estimated by the formula t 53. If x 81b2 and b 0, then x B A 9b. B 9b. C 3b27 . D 27b3. , where t is in seconds and s is in 2s 9.8 meters. If Julie jumps from an airplane, how long will it take her to free fall 750 meters? Surface area of a cube 6s2 about 12.4 s METEOROLOGY For Exercises 26 and 27, use the following information. To estimate how long a thunderstorm will last, meteorologists can use the formula t , where t is the time in hours and d is the diameter of the storm in miles. d3 216 26. A thunderstorm is 8 miles in diameter. Estimate how long the storm will last. Give your answer in simplified form and as a decimal. 83 h 1.5 h 9 27. Will a thunderstorm twice this diameter last twice as long? Explain. No; it will last about 4.4 h, or nearly 3 times as long. NAME ______________________________________________ DATE WEATHER For Exercises 54 and 55, use the following information. The formula y 91.4 (91.4 t) 0.478 0.301x 0.02 can be used to find the windchill factor. In this formula, y represents the windchill factor, t represents the air temperature in degrees Fahrenheit, and x represents the wind speed in miles per hour. Suppose the air temperature is 12°. Pre-Activity 54. Use a graphing calculator to find the wind speed to the nearest mile per hour if it feels like 9° with the windchill factor. 7 mph Reading the Lesson ____________ PERIOD _____ Reading 11-1 Readingto to Learn Learn Mathematics Mathematics, p. 647 Simplifying Radical Expressions ELL How are radical expressions used in space exploration? Read the introduction to Lesson 11-1 at the top of page 586 in your textbook. Suppose you want to calculate the escape velocity for a spacecraft taking off from the planet Mars. When you substitute numbers in the formula, which number is sure to be the same as in the calculation for the escape velocity for a spacecraft taking off from Earth? the value of G 1. a. How can you tell that the radical expression 28x2y4 is not in simplest form? The radicand contains perfect square factors other than 1. 55. What does it feel like to the nearest degree if the wind speed is 4 miles per hour? 6°F www.algebra1.com/self_check_quiz b. To simplify 28x2y4, you first find the You then apply the Lesson 11-1 Simplifying Radical Expressions 591 prime factorization Product Property of Square Roots 4 7 x2 y4 is equal to the product 4 again to get a final answer of 2 | x | y27 . of 28x2y4. . In this case, 7 x 2 y 4 . You can simplify 2. Why is it correct to write y4 y2, with no absolute value sign, but not correct to write x2 x? NAME ______________________________________________ DATE 11-1 Enrichment Enrichment, ____________ PERIOD _____ p. 648 Squares and Square Roots From a Graph The graph of y x2 can be used to find the squares and square roots of numbers. rationalizing the denominator 4. What should you do to write the conjugate of a binomial of the form ab cd ? To write the conjugate of a binomial of the form ab cd ? The arrows show that 32 9. A small part of the graph at y x2 is shown below. A 1:10 ratio for unit length on the y-axis to unit length on the x-axis is used. Example Find 11 . 3.3 The arrows show that 11 to the nearest tenth. y 12t 15 3. What method would you use to simplify ? y To find the square of 3, locate 3 on the x-axis. Then find its corresponding value on the y-axis. To find the square root of 4, first locate 4 on the y-axis. Then find its corresponding value on the x-axis. Following the arrows on the graph, you can see that 4 2. Sample answer: The square of y 2 is y 4 and the expressions y 4 and y 2 both represent positive numbers for all values of y. Although it is true 2 2 that the square of x is x , when x is less than 0, x represents a positive quantity and x represents a negative quantity. Change the plus sign to a minus sign; change the minus sign to a plus sign. O x Helping You Remember 5. What should you remember to check for when you want to determine if a radical expression is in simplest form? Sample answer: Check radicands for perfect squares and fractions, and check fractions for radicals in the denominator. 20 Lesson 11-1 Simplifying Radical Expressions 591 Lesson 11-1 Graphing Calculator Extending the Lesson 4 Assess 1 x 2 x. Using the properties of exponents, simplify each expression. 1 5 57. x 2 x2 1 1 56. x 2 x 2 x Open-Ended Assessment Writing Have students copy the Concept Summary on p. 589 into their Study Notebooks. For each of the three conditions listed, have students provide an example of an expression that does not satisfy the condition and then show how they simplified the expression to satisfy the condition. Radical expressions can be represented with fractional exponents. For example, 4 x2 x 3 58. x 2 or xx a 5 a . a 6 or ★ 59. Simplify the expression 3 6 aa a 1 for y. ★ 60. Solve the equation y3 3 3 ★ 61. Write 1 in simplest form. s2t 2 8 s5t4 3 3 s18t 6 s Maintain Your Skills Mixed Review Find the next three terms in each geometric sequence. 62. 2, 6, 18, 54 162, 486, 1458 64. 384, 192, 96, 48 24, 12, 6 3 3 3 3 3 3 66. 3, , , , , 4 16 64 256 1024 4096 Getting Ready for Lesson 11-2 (Lesson 10-7) 63. 1, 2, 4, 8 16, 32, 64 1 2 65. , , 4, 24 144, 864, 5184 9 3 67. 50, 10, 2, 0.4 0.08, 0.016, 0.0032 68. BIOLOGY A certain type of bacteria, if left alone, doubles its number every 2 hours. If there are 1000 bacteria at a certain point in time, how many bacteria will there be 24 hours later? (Lesson 10-6) 4,096,000 PREREQUISITE SKILL Students will learn how to perform operations with radical expressions in Lesson 11-2. In order to perform operations with radical expressions, students must be able to use the Distributive Property to multiply binomials. Use Exercises 84–89 to determine your students’ familiarity with multiplying binomials. 69. PHYSICS According to Newton’s Law of Cooling, the difference between the temperature of an object and its surroundings decreases in time exponentially. Suppose a cup of coffee is 95°C and it is in a room that is 20°C. The cooling of the coffee can be modeled by the equation y 75(0.875)t, where y is the temperature difference and t is the time in minutes. Find the temperature of the coffee after 15 minutes. (Lesson 10-6) 84.9°C Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 70. 6x2 7x 5 (3x 5)(2x 1) 71. 35x2 43x 12 (5x 4)(7x 3) 72. 5x2 3x 31 prime 73. 3x2 6x 105 3(x 7)(x 5) 74. 4x2 12x 15 prime 75. 8x2 10x 3 (4x 3)(2x 1) Find the solution set for each equation, given the replacement set. (Lesson 4-4) 76. y 3x 2; {(1, 5), (2, 6), (2, 2), (4, 10)} {(1, 5), (4, 10)} 77. 5x 2y 10; {(3, 5), (2, 0), (4, 2), (1, 2.5)} {(2, 0), (1, 2.5)} 78. 3a 2b 11; {(3, 10), (4, 1), (2, 2.5), (3, 2)} {(3, 10), (2, 2.5)} 3 1 79. 5 x 2y; (0, 1), (8, 2), 4, , (2, 1) 4, 1 , (2, 1) 2 2 2 Solve each equation. Then check your solution. (Lesson 3-3) 80. 40 5d 8 81. 20.4 3.4y 6 h 82. 25 275 11 Getting Ready for the Next Lesson PREREQUISITE SKILL Find each product. (To review multiplying binomials, see Lesson 8-7.) 84. (x 3)(x 2) x2 x 6 85. 11t 6 88. (5x 3y)(3x y) 15x2 4xy 3y2 86. (2t 1)(t 6) 592 592 Chapter 11 Radical Expressions and Triangles r 29 83. 65 1885 Chapter 11 Radical Expressions and Triangles 2t 2 (a 2)(a 5) a2 7a 10 87. (4x 3)(x 1) 4x2 x 3 89. (3a 2b)(4a 7b) 12a2 13ab 14b2 Operations with Radical Expressions Lesson Notes • Add and subtract radical expressions. 1 Focus • Multiply radical expressions. can you use radical expressions to determine how far a person can see? The formula d 3h represents 2 World’s Tall Structures the distance d in miles that a person h feet high can see. To determine how much farther a person can see from atop the Sears Tower than from atop the Empire State Building, we can substitute the heights of both buildings into the equation. 5-Minute Check Transparency 11-2 Use as a quiz or review of Lesson 11-1. Mathematical Background notes are available for this lesson on p. 584C. 984 feet 1250 feet Eiffel Empire State Tower, Building, Paris New York 1380 feet Jin Mau Building, Shanghai 1450 feet 1483 feet Sears Petronas Tower, Towers, Chicago Kuala Lumpur ADD AND SUBTRACT RADICAL EXPRESSIONS Radical expressions in which the radicands are alike can be added or subtracted in the same way that monomials are added or subtracted. Monomials Radical Expressions 2x 7x (2 7)x 211 711 (2 7)11 911 9x 152 32 (15 3)2 122 15y 3y (15 3)y 12y Notice that the Distributive Property was used to simplify each radical expression. Example 1 Expressions with Like Radicands Simplify each expression. a. 43 63 53 43 63 53 (4 6 5)3 53 Distributive Property Simplify. b. 125 37 67 85 Commutative 125 37 67 85 125 85 37 67 Property (12 8)5 (3 6)7 Distributive Property 45 97 can you use radical expressions to determine how far a person can see? Ask students: • What is an assumption that the formula makes? Sample answer: That there are no obstructions that would block a person from seeing the entire distance. • Assuming that nothing blocks the view, how much farther can a person atop the Sears Tower see than a person atop the Empire State Building? about 3.3 miles farther • Mountain Climbing Mount Everest, the tallest mountain in the world, is about 12,000 feet above the plateau out of which it rises. How far would a climber be able to see from the peak of Mount Everest? about 134 miles Simplify. In Example 1b, 45 97 cannot be simplified further because the radicands are different. There are no common factors, and each radicand is in simplest form. If the radicals in a radical expression are not in simplest form, simplify them first. Lesson 11-2 Operations with Radical Expressions 593 Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 649–650 • Skills Practice, p. 651 • Practice, p. 652 • Reading to Learn Mathematics, p. 653 • Enrichment, p. 654 • Assessment, p. 699 Parent and Student Study Guide Workbook, p. 84 Prerequisite Skills Workbook, pp. 37–38 Transparencies 5-Minute Check Transparency 11-2 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 593 Example 2 Expressions with Unlike Radicands 2 Teach Simplify 220 345 180 . ADD AND SUBTRACT RADICAL EXPRESSIONS In-Class Examples 220 22 5 3 32 5 62 5 345 180 2 2 22 5 3 32 5 62 5 Power Point® 225 335 65 1 Simplify each expression. 45 95 65 a. 65 25 55 35 195 b. 72 811 411 62 2 411 The simplified form is 195. 2 627 812 275 You can use a calculator to verify that a simplified radical expression is equivalent to the original expression. Consider Example 2. First, find a decimal approximation for the original expression. 443 KEYSTROKES: MULTIPLY RADICAL EXPRESSIONS In-Class Example 2 2nd [2 ] 20 ) 180 ) ENTER 3 2nd [2 ] 45 ) [2 ] 2nd 42.48529157 Next, find a decimal approximation for the simplified expression. Power Point® KEYSTROKES: 3 Find the area of a rectangle 19 2nd [2 ] 5 ENTER 42.48529157 Since the approximations are equal, the expressions are equivalent. with a width of 46 210 and a length of 53 75. 1830 102 MULTIPLY RADICAL EXPRESSIONS Multiplying two radical expressions with different radicands is similar to multiplying binomials. Example 3 Multiply Radical Expressions Find the area of the rectangle in simplest form. To find the area of the rectangle multiply the measures of the length and width. 45 23 36 10 45 23 36 10 First terms Inner terms Last terms Study Tip Outer terms 45 36 45 10 23 36 23 10 Look Back To review the FOIL method, see Lesson 8-7. 1230 450 618 230 Multiply. 226 222 1230 45 3 30 Prime factorization 1230 202 182 230 Simplify. 1430 382 Combine like terms. The area of the rectangle is 1430 382 square units. 594 Chapter 11 Radical Expressions and Triangles Differentiated Instruction Visual/Spatial Have students write all the perfect squares from 1 to 100 on an index card using a colored pen or pencil. Then have them rework Example 3. After they multiply the binomials, have students use their colored pen or pencil to circle the terms inside radicals that have perfect square factors. Using the same color, have students write the factorization in the line below each radical expression that can be factored. 594 Chapter 11 Radical Expressions and Triangles Concept Check 1. Explain why you should simplify each radical in a radical expression before adding or subtracting. to determine if there are any like radicands 2. Explain how you use the Distributive Property to simplify like radicands that are added or subtracted. See margin. 3. OPEN ENDED Choose values for x and y. Then find x y . 2 GUIDED PRACTICE KEY Exercises Examples 4, 5 6–9 10–13 1 2 3 Sample answer: 2 3 2 26 3 or 5 26 Simplify each expression. 4. 43 73 113 5. 26 76 56 6. 55 320 5 7. 23 12 43 8. 35 56 320 95 56 9. 83 3 9 93 3 Find each product. 10. 28 43 4 46 Applications 11. 4 5 3 5 17 75 12. GEOMETRY Find the perimeter and the area of a square whose sides measure 4 36 feet. P 16 126 ft; A 70 246 ft2 13. ELECTRICITY The voltage V required for a circuit is given by V PR , where P is the power in watts and R is the resistance in ohms. How many more volts are needed to light a 100-watt bulb than a 75-watt bulb if the resistance for both is 110 ohms? 10110 5330 14.05 volts ★ indicates increased difficulty Practice and Apply Homework Help For Exercises See Examples 14–21 22–29 30–48 1 2 3 Simplify each expression. 14. 85 35 115 16. 215 615 315 715 15. 36 106 136 17. 519 619 1119 0 18. 16x 2x 18x 19. 35b 45b 115b 105b Extra Practice 20. 83 22 32 53 21. 46 17 62 417 20. 133 2 22. 18 12 8 52 23 23. 6 23 12 24. 37 228 7 See page 844. 25. 6 43 250 332 22 32 410 21. 46 62 ★ 26. 2 1 ★ 27. 10 25 2 2 5 5 17 537 ★ 28. 33 45 3 13 43 35 ★ 29. 6 74 328 10 17 About the Exercises … Organization by Objective • Add and Subtract Radical Expressions: 14–29 • Multiply Radical Expressions: 30–37 Odd/Even Assignments Exercises 14–39 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 42 requires the use of the Internet or other research materials. Assignment Guide 7 Find each product. 34. 103 16 Study Notebook Have students— • include examples of how to add, subtract, and multiply radical expressions. • include any other item(s) that they find helpful in mastering the skills in this lesson. 2 Guided Practice 3 Practice/Apply 30. 63 52 32 103 31. 5210 32 102 310 32. 3 5 3 5 4 33. 7 10 2 59 1410 ★ 34. 6 8 24 2 ★ 35. 5 2 14 35 37 ★ 36. 210 315 33 22 195 ★ 37. 52 35 210 3 152 115 Basic: 15–25 odd, 31, 33, 39, 41, 42, 49–76 Average: 15–39 odd, 41–44, 49–76 Advanced: 14–40 even, 45–70 (optional: 71–76) 38. GEOMETRY Find the perimeter of a rectangle whose length is 87 45 inches and whose width is 27 35 inches. 20 7 25 in. www.algebra1.com/extra_examples Lesson 11-2 Operations with Radical Expressions 595 Answer 2. The Distributive Property allows you to add like terms. Radicals with like radicands can be added or subtracted. Lesson 11-2 Operations with Radical Expressions 595 NAME ______________________________________________ DATE 39. GEOMETRY The perimeter of a rectangle is 23 411 6 centimeters, ____________ PERIOD _____ Study Guide andIntervention Intervention, 11-2 Study Guide and p. 649 (shown) and p. 650 Operations with Radical Expressions and its length is 211 1 centimeters. Find the width. Add and Subtract Radical Expressions When adding or subtracting radical expressions, use the Associative and Distributive Properties to simplify the expressions. If radical expressions are not in simplest form, simplify them. Example 1 Simplify 106 53 63 46 . 53 63 46 (10 4)6 (5 6)3 106 6 6 3 Example 2 Simplify. 1 2 Simplify. the diagonals of the rhombus. What is the area of the rhombus at the right? 156 cm2 Simplify. Lesson 11-2 Distributive Property Exercises 1. 25 45 65 2. 6 46 36 2 3. 8 2 5 6. 23 6 53 33 7. 12 23 53 3 6 41. How much farther can a person see from atop the Sears Tower than from atop the Empire State Building? 587 253 3.34 mi 8. 3 6 3 2 50 24 5 6 2 2 9. 8a 2a 52a 62a 10. 54 24 56 43 3 1 3 12. 12 176 6 1 6 42. A person atop the Empire State Building can see approximately 4.57 miles farther than a person atop the Texas Commerce Tower in Houston. Explain how you could find the height of the Texas Commerce Tower. See margin. 73 3 1 3 14. 80 20 180 85 83 125 3 1 3 15. 50 18 75 27 8 2 2 3 16. 23 445 2 17. 125 2 1 5 1 235 3 5 3 3 18. 1 6 12 3 2 33 4 3 283 12 1. 830 430 430 2. 25 75 55 45 3. 713x 1413x 213x 513x 4. 245 420 145 5. 40 10 90 410 6. 232 350 318 142 7. 27 18 300 32 133 8. 58 320 32 62 65 Source: www.the-skydeck.com NAME ______________________________________________ DATE ____________ PERIOD _____ p. 651 and Practice, (shown) Operationsp. with652 Radical Expressions Simplify each expression. 9. 14 614 7 2 7 10. 50 32 11. 519 428 819 63 Online Research Data Update What are the tallest buildings and towers in the world today? Visit www.algebra1.com/data_update to learn more. Distance The Sears Tower was the tallest building in the world from 1974 to 1996. You can see four states from its roof: Michigan, Indiana, Illinois, and Wisconsin. Skills Practice, 11-2 Practice (Average) ENGINEERING For Exercises 43 and 44, use the following information. rate F of water passing through it in gallons per minute. 43. Find the radius of a pipe that can carry 500 gallons of water per minute. Round to the nearest whole number. 6 in. 172 2 1 2 10 33 44. An engineer determines that a drainpipe must be able to carry 1000 gallons of water per minute and instructs the builder to use an 8-inch radius pipe. Can the builder use two 4-inch radius pipes instead? Justify your answer. Find each product. (10 15 ) 215 310 13. 6 14. 5 (52 48 ) 310 15. 27 (312 58 ) 1221 2014 16. (5 15 ) 40 1015 2 17. (10 6 )(30 18 ) 43 18. (8 12 )(48 18 ) 36 146 19. (2 28 )(36 5 ) 20. (43 25 )(310 56 ) 303 510 F relates the radius r of a drainpipe in inches to the flow 5 The equation r 12. 310 75 240 412 319 117 54 cm DISTANCE For Exercises 41 and 42, refer to the application at the beginning of the lesson. 4. 375 25 153 25 5. 20 25 35 36 cm A d1d2, where d1 and d2 are the lengths of Simplify. Simplify each expression. 13. 54 40. GEOMETRY A formula for the area A of a rhombus can be found using the formula Associative and Distributive Properties Simplify 312 575 . 575 3 22 3 5 52 3 312 3 23 5 53 6 3 253 31 3 11. 3 3 2 cm No, each pipe would need to carry 500 gallons per minute, so the pipes would need at least a 6-inch radius. 230 302 MOTION For Exercises 45–47, use the following information. The velocity of an object dropped from a certain height can be found using the SOUND For Exercises 21 and 22, use the following information. t 27 3, The speed of sound V in meters per second near Earth’s surface is given by V 20 where t is the surface temperature in degrees Celsius. formula v 2gd , where v is the velocity in feet per second, g is the acceleration due to gravity, and d is the distance in feet the object drops. 21. What is the speed of sound near Earth’s surface at 15°C and at 2°C in simplest form? 2402 m/s, 10011 m/s 22. How much faster is the speed of sound at 15°C than at 2°C? 45. Find the speed of an object that has fallen 25 feet and the speed of an object that has fallen 100 feet. Use 32 feet per second squared for g. 40 ft/s; 80 ft/s 2402 10011 7.75 m/s GEOMETRY For Exercises 23 and 24, use the following information. A rectangle is 57 23 centimeters long and 67 33 centimeters wide. 23 cm 23. Find the perimeter of the rectangle in simplest form. 227 46. When you increased the distance by 4 times, what happened to the velocity? 24. Find the area of the rectangle in simplest form. 192 321 cm2 NAME ______________________________________________ DATE The velocity doubled. 47. MAKE A CONJECTURE Estimate the velocity of an object that has fallen 225 feet. Then use the formula to verify your answer. The velocity should be ____________ PERIOD _____ Reading 11-2 Readingto to Learn Learn MathematicsELL Mathematics, p. 653 Operations with Radical Expressions Pre-Activity 9 or 3 times the velocity of an object falling 25 feet; 3 40 120 ft/s, 25) 120 ft/s. 2(32)(2 How can you use radical expressions to determine how far a person can see? Read the introduction to Lesson 11-2 at the top of page 593 in your textbook. 48. WATER SUPPLY The relationship between a city’s size and its capacity to supply Suppose you substitute the heights of the Sears Tower and the Empire State Building into the formula to find how far you can see from atop each building. What operation should you then use to determine how much farther you can see from the Sears Tower than from the Empire State Building? water to its citizens can be described by the expression 1020P 1 0.01P , where P is the population in thousands and the result is the number of gallons per minute required. If a city has a population of 55,000 people, how many gallons per minute must the city’s pumping station be able to supply? subtraction Reading the Lesson about 7003.5 gal/min 1. Indicate whether the following expressions are in simplest form. Explain your answer. 596 Chapter 11 Radical Expressions and Triangles a. 63 12 No; 12 can be simplified to 22 3 or 23 . b. 126 710 Yes; both the addends are radical expressions in simplest form, the radicands are different, and there are no common factors. 2. Below the words First terms, Outer terms, Inner terms, and Last terms, write the products you would use to simplify the expression (215 315 )(63 52 ). First terms Outer terms Inner terms (215 )(63 ) (215 )(52 ) (35 )(63) Last terms (35 )(52 ) Helping You Remember 3. How can you use what you know about adding and subtracting monomials to help you remember how to add and subtract radical expressions? Sample answer: Check that the addends have been simplified. Next, group addends that involve like radicals. Then use the Distributive Property to combine the addends that involve like radicals. 596 NAME ______________________________________________ DATE 11-2 Enrichment Enrichment, ____________ PERIOD _____ p. 654 42. Approximately 1000 feet; determine h so that The Wheel of Theodorus The Greek mathematicians were intrigued by problems of representing different numbers and expressions using geometric constructions. 1 D C 1 Theodorus, a Greek philosopher who lived about 425 B.C., is said to have discovered a way to construct the sequence 1, 2, 3, 4, … . B 1 The beginning of his construction is shown. You start with an isosceles right triangle with sides 1 unit long. O 1 Use the figure above. Write each length as a radical expression in simplest form. 1. line segment AO 1 2. line segment BO 2 3. line segment CO 3 4. line segment DO 4 Chapter 11 Radical Expressions and Triangles Answer A 3(1250) 3h 4.57; 2 2 may use guess and test, graphical, or analytical methods. 49. CRITICAL THINKING Find a counterexample to disprove the following statement. Sample answer: a 4, b 9: 4 9 4 9 4 Assess For any numbers a and b, where a 0 and b 0, a b a b. a b a b 50. CRITICAL THINKING Under what conditions is true? a 0 or b 0 or both 2 51. WRITING IN MATH 2 2 Answer the question that was posed at the beginning of the lesson. See margin. How can you use radical expressions to determine how far a person can see? Include the following in your answer: • an explanation of how this information could help determine how far apart lifeguard towers should be on a beach, and • an example of a real-life situation where a lookout position is placed at a high point above the ground. Standardized Test Practice C 7 57 B 47 D 77 PREREQUISITE SKILL Students will learn how to solve radical equations in Lesson 11-3. In order to solve radical equations, students will need to be able to recognize and find special products. Use Exercises 71–76 to determine your students’ familiarity with finding special products. 53. Simplify 34 12 . D 2 A 43 6 B 283 C 28 163 D 48 283 Maintain Your Skills Mixed Review 56. 14xy y Simplify. (Lesson 11-1) 54. 40 210 55. 128 82 5 50 57. 8 2 58. Find the nth term of each geometric sequence. 60. a1 4, n 6, r 4 2 y3 56. 196x 225c4d 5c2d 18c2 2 59. 5103 4096 Assessment Options 314 63a 128a3b2 16ab Quiz (Lessons 11-1 and 11-2) is available on p. 699 of the Chapter 11 Resource Masters. (Lesson 10-7) 61. a1 7, n 4, r 9 62. a1 2, n 8, r 0.8 0.4194304 Solve each equation by factoring. Check your solutions. (Lesson 9-5) 6 9 36 63. 81 49y2 64. q2 0 121 11 7 5 5 65. 48n3 75n 0 , 0, 66. 5x3 80x 240 15x2 {4, 3, 4} 4 4 Solve each inequality. Then check your solution. (Lesson 6-2) 5 3 w 7k 21 67. 8n 5 n 68. 14 w 126 69. k 9 2 10 8 5 70. PROBABILITY A student rolls a die three times. What is the probability that each roll is a 1? (Lesson 2-6) 1 216 Getting Ready for the Next Lesson PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.) 71. (x 2)2 x2 4x 4 72. (x 5)2 x2 10x 25 73. (x 6)2 x2 12x 36 74. (3x 1)2 9x2 6x 1 75. (2x 3)2 www.algebra1.com/self_check_quiz 4x2 12x 9 Speaking Ask students to compare and contrast adding, subtracting, and multiplying radical expressions and variable expressions. What are the similarities? What are the differences? Allow students to give examples on the chalkboard if they wish. Getting Ready for Lesson 11-3 52. Find the difference of 97 and 228 . C A Open-Ended Assessment 76. (4x 7)2 16x2 56x 49 Lesson 11-2 Operations with Radical Expressions 597 Answer 51. The distance a person can see is related to the height of the person using d 3h2 . Answers should include the following. • You can find how far each lifeguard can see from the height of the lifeguard tower. Each tower should have some overlap to cover the entire beach area. • On early ships, a lookout position (Crow’s nest) was situated high on the foremast. Sailors could see farther from this position than from the ship’s deck. Lesson 11-2 Operations with Radical Expressions 597 Lesson Notes Radical Equations • Solve radical equations. 1 Focus • Solve radical equations with extraneous solutions. 5-Minute Check Transparency 11-3 Use as a quiz or review of Lesson 11-2. Vocabulary • radical equation • extraneous solution Mathematical Background notes are available for this lesson on p. 584D. are radical equations used to find freefall times? Ask students: h 4 are radical equations used to find free-fall times? Skydivers fall 1050 to 1480 feet every 5 seconds, reaching speeds of 120 to 150 miles per hour at terminal velocity. It is the highest speed they can reach and occurs when the air resistance equals the force of gravity. With no air resistance, the time t in seconds that it takes an object to fall h h feet can be determined by the equation t . 4 How would you find the value of h if you are given the value of t? h that contain radicals with RADICAL EQUATIONS Equations like t 4 • Is in simplest form? variables in the radicand are called radical equations. To solve these equations, first isolate the radical on one side of the equation. Then square each side of the equation to eliminate the radical. Explain. Yes, since there is no radical in the denominator, the expression is in simplest form. x 4 • How would you solve t Example 1 Radical Equation with a Variable for x? Multiply each side by 4. • What do you think might be a way to remove the radical sign from 4t h? Square each side of the equation. FREE-FALL HEIGHT Two objects are dropped simultaneously. The first object reaches the ground in 2.5 seconds, and the second object reaches the ground 1.5 seconds later. From what heights were the two objects dropped? Find the height of the first object. Replace t with 2.5 seconds. h t Original equation 4 h 2.5 Replace t with 2.5. 4 10 h Multiply each side by 4. 102 h 100 h 2 Square each side. Simplify. h CHECK t 4 100 t 4 10 t 4 t 2.5 Original equation h 100 100 10 Simplify. The first object was dropped from 100 feet. 598 Chapter 11 Radical Expressions and Triangles Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 655–656 • Skills Practice, p. 657 • Practice, p. 658 • Reading to Learn Mathematics, p. 659 • Enrichment, p. 660 Graphing Calculator and Spreadsheet Masters, p. 43 Parent and Student Study Guide Workbook, p. 85 Transparencies 5-Minute Check Transparency 11-3 Answer Key Transparencies Technology Interactive Chalkboard The time it took the second object to fall was 2.5 1.5 seconds or 4 seconds. h t Original equation h 4 Replace t with 4. 4 4 16 h 2 Teach RADICAL EQUATIONS In-Class Examples Multiply each side by 4. 162 h 256 h 2 Power Point® Square each side. 1 FREE-FALL HEIGHT An object Simplify. is dropped from an unknown height and reaches the ground in 5 seconds. From what height is it dropped? The object is dropped from 400 ft. The second object was dropped from 256 feet. Check this solution. Example 2 Radical Equation with an Expression Solve x 1 7 10. 1 7 10 x 13 x 2 x 1 32 x19 x8 Teaching Tip Remind students that they must isolate the radical part of the expression before squaring each side. Original equation Subtract 7 from each side. Square each side. 2 Solve x 3 8 15. 52 2 x 1 x 1 Subtract 1 from each side. The solution is 8. Check this result. EXTRANEOUS SOLUTIONS EXTRANEOUS SOLUTIONS Squaring each side of an equation sometimes produces extraneous solutions. An extraneous solution is a solution derived from an equation that is not a solution of the original equation. Therefore, you must check all solutions in the original equation when you solve radical equations. Example 3 Variable on Each Side Solve x 2 x 4. x2x4 2 x 2 (x 4)2 Study Tip Look Back To review Zero Product Property, see Lesson 9-2. Original equation Square each side. x2 x2 8x 16 Simplify. 0 x2 9x 14 Subtract x and 2 from each side. 0 (x 7)(x 2) Factor. x70 or x 2 0 x7 x2 Zero Product Property 33 ⻫ In Chapter 10, students learned that there are two solutions to most quadratic equations. In this lesson, students will learn that when solving radical equations, it is possible to introduce a second solution when squaring both sides of an equation. However, since the original equation was not quadratic, one of the solutions will be extraneous. Solve. CHECK x2x4 274 7 9 3 Building on Prior Knowledge x7 x2x4 2224 4 2 2 2 ⫻ In-Class Example Power Point® 3 Solve 2 y y. 1 x2 Since 2 does not satisfy the original equation, 7 is the only solution. www.algebra1.com/extra_examples Lesson 11-3 Radical Equations Differentiated Instruction 599 ELL Verbal/Linguistic Have students write a short paragraph in their own words explaining why checking solutions is important when solving radical equations. Students should include an example of a radical equation that has an extraneous solution. Lesson 11-3 Radical Equations 599 3 Practice/Apply Study Notebook Have students— • include examples of how to solve radical equations. • include any other item(s) that they find helpful in mastering the skills in this lesson. Solving Radical Equations 2. (10, 5) 3. x 10; the solution is the same as the solution from the graph. However, when solving algebraically, you have to check that x 3 is an extraneous solution. Concept Check FIND THE ERROR This problem may require close inspection for students to find the error. Tell students to focus on checking the given solutions. One solution produces an impossible radicand. Also point out that Alex and Victor could have begun by multiplying each side by 1 to eliminate the negative signs. 1. Isolate the radical on one side of the equation. Square each side of the equation and simplify. Then check for extraneous solutions. 4. Alex; the square of x 5 is x 5. You can use a TI-83 Plus graphing calculator to solve radical equations 3x 5 x 5. Clear the Y list. Enter the left side of the such as 3x 5. Enter the right side of the equation as equation as Y1 Y2 x 5. Press GRAPH . Think and Discuss 1. Sketch what is shown on the screen. See margin. 2. Use the intersect feature on the CALC menu, to find the point of intersection. 3. Solve the radical equation algebraically. How does your solution compare to the solution from the graph? 1. Describe the steps needed to solve a radical equation. 2. Explain why it is necessary to check for extraneous solutions in radical equations. The solution may not satisfy the original equation. 3. OPEN ENDED Give an example of a radical equation. Then solve the equation for the variable. Sample answer: x 1 8; 63 4. FIND THE ERROR Alex and Victor are solving x 5 2. Alex Victor x – 5 = –2 –∫ –∂ x – 5 = –2 x – 5)2 = (–2)2 (– ∫ (–∂ x – 5) 2 = (–2) 2 x–5=4 –(x – 5) = 4 x=9 –x + 5 = 4 x = 1 Answer Who is correct? Explain your reasoning. 1. Guided Practice Solve each equation. Check your solution. 5. x 5 25 6. 2b 8 no solution 7. 7x 7 7 8. 3a 6 12 9. 8s 15 2 11. 5x 1 2 6 3 Application GUIDED PRACTICE KEY Exercises Examples 5–8, 14–16 9–11 12, 13 1 600 2 3 12. 3x 5 x 5 10 10. 7x 18 9 9 13. 4 x2x 6 OCEANS For Exercises 14–16, use the following information. Tsunamis, or large tidal waves, are generated by undersea earthquakes in the Pacific d, where d is the Ocean. The speed of the tsunami in meters per second is s 3.1 depth of the ocean in meters. 15. 5994 m 14. Find the speed of the tsunami if the depth of the water is 10 meters. 9.8 m/s 15. Find the depth of the water if a tsunami’s speed is 240 meters per second. 16. A tsunami may begin as a 2-foot high wave traveling 450–500 miles per hour. It can approach a coastline as a 50-foot wave. How much speed does the wave lose if it travels from a depth of 10,000 meters to a depth of 20 meters? Chapter 11 Radical Expressions and Triangles approximately 296 m/s Solving Radical Equations An alternative to graphing each side of the equation separately is to graph the single equation as it is. Move the x 5 to the left side of the equation and then enter the equation as Y1 3x 5 x 5. Then press 2nd [CALC] 2 to calculate the zero point of the graph. The zero point is the solution, which for this equation is 10. 600 Chapter 11 Radical Expressions and Triangles ★ indicates increased difficulty Practice and Apply Homework Help For Exercises See Examples 17–34 35–47 48–59 1, 2 3 1–3 Extra Practice See page 844. About the Exercises … Solve each equation. Check your solution. 17. a 10 100 18. k 4 16 19. 52 x 50 20. 37 y 63 21. 34a 2 10 4 22. 3 5n 18 9 23. x 3 5 no solution 24. x 5 26 29 25. 3x 12 33 5 26. 2c 4 8 34 27. 4b 1 3 0 2 28. 3r 5 7 3 no solution 3 4t 30. 5 2 0 3 25 32. y 2 y2 5 y4 0 29. 4x 9 3 5 180 31. x2 9 x 14 x 4 2 Organization by Objective • Radical Equations: 17–34 • Extraneous Solutions: 35–47 Odd/Even Assignments Exercises 17–46 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide 33. The square root of the sum of a number and 7 is 8. Find the number. 57 Basic: 17–47 odd, 48, 49, 60–63, 70–89 Average: 17–47 odd, 48–53, 60–63, 70–89 (optional: 64–69) Advanced: 18–46 even, 50–85 (optional: 86–89) All: Practice Quiz 1 (1–10) 34. The square root of the quotient of a number and 6 is 9. Find the number. 486 Solve each equation. Check your solution. 35. x 6x 2 36. x x 20 5 37. 5x 6 x 2, 3 38. 28 3x x 4 39. x1x1 3 40. 1 2b 1 b 0 41. 4 m2m 6 42. 3d 8 d 2 3, 4 43. x 6x4 2 44. 6 3x x 16 10 45. 121 r 11 46. 5p2 7 2p 2r2 7 47. State whether the following equation is sometimes, always, or never true. 5 )2 x 5 sometimes (x AVIATION For Exercises 48 and 49, use the following information. The formula L kP represents the relationship between a plane’s length L and the pounds P its wings can lift, where k is a constant of proportionality calculated for a plane. Aviation Piloted by A. Scott Crossfield on November 20, 1953, the Douglas D-558-2 Skyrocket became the first aircraft to fly faster than Mach 2, twice the speed of sound. Source: National Air and Space Museum 48. The length of the Douglas D-558-II, called the Skyrocket, was approximately 42 feet, and its constant of proportionality was k 0.1669. Calculate the maximum takeoff weight of the Skyrocket. 10,569 lb 49. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds. Determine the value of k for this plane. 0.0619 GEOMETRY For Exercises 50–53, use the figure below. The area A of a circle is equal to r2 where r is the radius of the circle. A 50. Write an equation for r in terms of A. r 51. The area of the larger circle is 96 square meters. Find the radius. 46 or 9.8 m 52. The area of the smaller circle is 48 square meters. Find the radius. 43 or 6.9 m ★ 53. If the area of a circle is doubled, what is the change in the radius? It increases by a factor of 2 . Lesson 11-3 Radical Equations 601 Lesson 11-3 Radical Equations 601 NAME ______________________________________________ DATE PHYSICAL SCIENCE For Exercises 54–56, use the following information. ____________ PERIOD _____ Study Guide andIntervention Intervention, 11-3 Study Guide and 32 ᐉ The formula P 2 gives the period of a pendulum of length ᐉ feet. The period p. 655 and p. 656 Radical(shown) Equations Radical Equations Equations containing radicals with variables in the radicand are called radical equations. These can be solved by first using the following steps. P is the number of seconds it takes for the pendulum to swing back and forth once. Isolate the radical on one side of the equation. Square each side of the equation to eliminate the radical. Step 1 Step 2 Example 1 x 16 2 x 2(16) 2 2 32 x (32)2 (x )2 1024 x x 2 Example 2 Solve 16 for x. Original equation Multiply each side by 2. Simplify. Square each side. Simplify. The solution is 1024, which checks in the original equation. 54. Suppose we want a pendulum to complete three periods in 2 seconds. How long should the pendulum be? 0.36 ft Solve 4x 7 2 7. 4x 7 2 7 4x 7 2 2 7 2 4x 7 5 2 ( 4x 7 ) 52 4x 7 25 4x 7 7 25 7 4x 32 x8 Original equation Subtract 2 from each side. Simplify. Square each side. Simplify. Add 7 to each side. Simplify. Divide each side by 4. The solution is 8, which checks in the original equation. Exercises 1. a 8 64 2. a 6 32 676 3. 2x 8 16 4. 7 26 n 23 5. a 6 36 6. 3r2 3 3 7. 23 y 12 8. 23a 2 7 6 3 4 10. 2c 3 5 11 9. x 4 6 40 11. 3b 2 19 24 9 10 3 3r 2 23 13. 1 2 4 3 16. 6x2 5x 2 , 14. 17. 6 8 12 x 2 1 1 2 2 5 2 12. 4x 1 3 15. 4 128 x 8 3 11 2623 x 3 18. 2 NAME ______________________________________________ DATE Skills Practice, 11-3 Practice (Average) 3x 5 ____________ PERIOD _____ p. 657 and Practice, p. 658 (shown) Radical Equations Lesson 11-3 Solve each equation. Check your solution. Physical Science The Foucault pendulum was invented in 1851. It demonstrates the rotation of Earth. The pendulum appears to change its path during the day, but it moves in a straight line. The path under the pendulum changes because Earth is rotating beneath it. Source: California Academy of Sciences 55. Two clocks have pendulums of different lengths. The first clock requires 1 second for its pendulum to complete one period. The second clock requires 2 seconds for its pendulum to complete one period. How much longer is one pendulum than the other? 2.43 ft 24t 2 ★ 56. Repeat Exercise 55 if the pendulum periods are t and 2t seconds. 2 SOUND For Exercises 57–59, use the following information. The speed of sound V near Earth’s surface can be found using the equation t 273, where t is the surface temperature in degrees Celsius. V 20 57. Find the temperature if the speed of sound V is 356 meters per second. 43.8°C 58. The speed of sound at Earth’s surface is often given at 340 meters per second, but that is only accurate at a certain temperature. On what temperature is this figure based? 16°C ★ 59. What is the speed of sound when the surface temperature is below 0°C? V 330.45 m/s 60. CRITICAL THINKING Solve h 9 h 3 . 3 Solve each equation. Check your solution. 1. b 8 64 2. 43 x 48 3. 24c 3 11 4 4. 6 2y 2 32 5. k 2 3 7 98 6. m 5 43 53 7. 6t 12 86 62 8. 3j 11 2 9 20 9. 2x 15 5 18 77 3 0 14 11. 6 61. WRITING IN MATH How are radical equations used to find free-fall times? 4 2 60 12. 6 2 no solution 10. 3s 5 Include the following in your answer: • the time it would take a skydiver to fall 10,000 feet if he falls 1200 feet every h 5 seconds and the time using the equation t , with an explanation of 4 why the two methods find different times, and 5r 6 3x 3 13. y y6 3 14. 15 2x x 3 15. w 4 w 4 4, 3 16. 17 k k5 8 17. 5m 16 m 2 4, 5 18. 24 8q q 3 3, 5 19. 4s 17 s30 2 20. 4 3m 28 m 1 21. 10p 61 7 p 6, 2 22. 2x2 9x 3 • ways that a skydiver can increase or decrease his speed. ELECTRICITY For Exercises 23 and 24, use the following information. The voltage V in a circuit is given by V PR , where P is the power in watts and R is the resistance in ohms. Standardized Test Practice 23. If the voltage in a circuit is 120 volts and the circuit produces 1500 watts of power, what is the resistance in the circuit? 9.6 ohms 24. Suppose an electrician designs a circuit with 110 volts and a resistance of 10 ohms. How much power will the circuit produce? 1210 watts QUANTITATIVE COMPARISON In Exercises 62 and 63, compare the quantity in Column A and the quantity in Column B. Then determine whether: FREE FALL For Exercises 25 and 26, use the following information. Assuming no air resistance, the time t in seconds that it takes an object to fall h feet can be h determined by the equation t . 4 25. If a skydiver jumps from an airplane and free falls for 10 seconds before opening the parachute, how many feet does the skydiver fall? 1600 ft 26. Suppose a second skydiver jumps and free falls for 6 seconds. How many feet does the second skydiver fall? 576 ft NAME ______________________________________________ DATE Answer the question that was posed at the beginning of the lesson. See margin. A the quantity in Column A is greater, B the quantity in Column B is greater, C both quantities are equal, or D the relationship cannot be determined from the given information. ____________ PERIOD _____ Reading 11-3 Readingto to Learn Learn Mathematics Mathematics, p. 659 Radical Equations Pre-Activity ELL Column A Column B the solution of the solution of x36 y 3 6 2 a 1 1 )2 (a How are radical equations used to find free-fall times? Read the introduction to Lesson 11-3 at the top of page 598 in your textbook. on one side of the equation? How can you isolate h 62. Multiply each side of the equation by 4. Reading the Lesson A 1. To solve a radical equation, you first isolate the radical on one side of the equation. Why do you then square each side of the equation? to eliminate the radical 63. 2. a. Provide the reason for each step in the solution of the given radical equation. 5x 1 4 x 3 5x 1 x 1 2 ( 5x 1) (x 1)2 5x 1 x2 2x 1 0 x2 3x 2 0 (x 1)(x 2) x 1 0 or x1 x20 x2 Original equation Add 4 to each side. Square each side. Simplify. Subtract 5x and add 1 to each side. Factor. Zero Product Property Solve. b. To be sure that 1 and 2 are the correct solutions, into which equation should you substitute to check? the original equation 3. a. How do you determine whether an equation has an extraneous solution? Substitute the solution(s) into the original equation. If a solution does not satisfy the original equation, then it is an extraneous solution. b. Is it necessary to check all solutions to eliminate extraneous solutions? Explain. Yes; since you square each side of a radical equation, and squaring each side can sometimes produce an extraneous solution, you need to check all solutions. The only way to be sure that a solution is not extraneous is to check it in the original equation. Helping You Remember 4. How can you use the letters ISC to remember the three steps in solving a radical equation? Sample answer: Isolate the radical on one side of the equation, Square each side to eliminate the radical, and Check for extraneous solutions. 602 602 C Chapter 11 Radical Expressions and Triangles NAME ______________________________________________ DATE 11-3 Enrichment Enrichment, ____________ PERIOD _____ p. 660 Special Polynomial Products Sometimes the product of two polynomials can be found readily with the use of one of the special products of binomials. For example, you can find the square of a trinomial by recalling the square of a binomial. Example 1 Find (x y z)2. (a b)2 a2 2 a b b2 [(x y) z]2 (x y)2 2(x y)z z2 x2 2xy y2 2xz 2yz z2 Example 2 Find (3t x 1)(3t x 1). (Hint: (3t x 1)(3t x 1) is the product of a sum 3t (x 1) and a difference 3t (x 1).) (3t x 1)(3t x 1) [3t (x 1)][3t (x 1)] 9t2 (x 1)2 9t2 x2 2x 1 Chapter 11 Radical Expressions and Triangles Graphing Calculator RADICAL EQUATIONS Use a graphing calculator to solve each radical equation. Round to the nearest hundredth. 65. 3x 8 5 11 66. x 6 4 x 2 67. 4x 5 x 7 15.08 Open-Ended Assessment 68. x 7 x 4 1.70 69. 3x 9 2x 6 no solution Modeling Write radical equations on the overhead projector, using a cut-out overlay for the radical. Have students explain what must be done to the equation in order to “remove” the radical. When students explain the correct procedure, remove the overlay and solve the equation. Maintain Your Skills Mixed Review Simplify each expression. (Lesson 11-2) 70. 56 126 176 Simplify. 71. 12 627 203 72. 18 52 332 42 (Lesson 11-1) 73. 192 83 74. 6 10 215 21 75. 10 3 310 3 Getting Ready for Lesson 11-4 Determine whether each trinomial is a perfect square trinomial. If so, factor it. (Lesson 9-6) 76. d2 50d 225 no 77. 4n2 28n 49 78. 16b2 56bc 49c2 yes; (2n 7)2 yes; (4b 7c)2 Find each product. (Lesson 8-7) 79. (r 3)(r 4) r 2 r 12 80. (3z 7)(2z 10) 6z2 44z 70 81. (2p 5)(3p2 4p 9) 6p3 7p 2 2p 45 82. PHYSICAL SCIENCE A European-made hot tub is advertised to have a temperature of 35°C to 40°C, inclusive. What is the temperature range for the 9 5 hot tub in degrees Fahrenheit? Use F C 32. (Lesson 6-4) 95° F 104° Write each equation in standard form. (Lesson 5-5) 3 7 83. y 2x 14x 7y 3 Getting Ready for the Next Lesson 84. y 3 2(x 6) 2x y 15 15x 2y 49 86. a 3, b 4 5 2 87. a 24, b 7 25 89. a 8, b 12 413 Lessons 11-1 through 11-3 (Lesson 11-1) 1. 48 43 2. 3 6 32 2 10 3 3. 2 2 10 5. 23 912 203 2 6. 3 6 15 66 115 311 7. GEOMETRY Find the area of a square whose side measure is 2 7 centimeters. 11 47 or 21.6 cm2 (Lesson 11-2) Solve each equation. Check your solution. (Lesson 11-3) 8. 15 x 4 1 9. 3x2 32 x 4 www.algebra1.com/self_check_quiz 10. 2x 1 2x 7 5 Lesson 11-3 Radical Equations Practice Quiz 1 The quiz provides students with a brief review of the concepts and skills in Lessons 11-1 through 11-3. Lesson numbers are given to the right of exercises or instruction lines so students can review concepts not yet mastered. Answer (Lesson 11-2) 4. 65 311 55 PREREQUISITE SKILL Students will learn about the Pythagorean Theorem in Lesson 11-4. In order to use the Pythagorean Theorem, students must be able to evaluate radical expressions for given values. Use Exercises 86–89 to determine your students’ familiarity with evaluating radical expressions. Assessment Options (To review evaluating expressions, see Lesson 1-2.) P ractice Quiz 1 Simplify. 85. y 2 7.5(x 3) PREREQUISITE SKILL Evaluate a2 b2 for each value of a and b. 88. a 1, b 1 Simplify. 4 Assess 64. 3 2x 7 8 603 61. You can determine the time it takes an object to fall from a given height using a radical equation. Answers should include the following. • It would take a skydiver approximately 42 seconds to fall 10,000 feet. Using the equation, it would take 25 seconds. The time is different in the two calculations because air resistance slows the skydiver. • A skydiver can increase the speed of his fall by lowering air resistance. This can be done by pulling his arms and legs close to his body. A skydiver can decrease his speed by holding his arms and legs out, which increases the air resistance. Lesson 11-3 Radical Equations 603 Graphing Calculator Investigation A Follow-Up of Lesson 11-3 A Follow-Up of Lesson 11-3 Graphs of Radical Equations Getting Started In order for a square root to be a real number, the radicand cannot be negative. When graphing a radical equation, determine when the radicand would be negative and exclude those values from the domain. Know Your Calculator The ZoomFit option in the ZOOM menu will let the calculator automatically zoom the viewing window to fit the graph. Suggest that students use this option with the examples in this investigation to get a better view of the shape of the graph of a radical equation. Example 1 Graph y x . State the domain of the graph. Enter the equation in the Y= list. KEYSTROKES: 2nd [2 ] X,T,,n ) GRAPH From the graph, you can see that the domain of x is {xx 0}. Teach [10, 10] scl: 1 by [10, 10] scl: 1 • Have students use the CALC Value function to find the value of the function at different x-values. Press 2nd [CALC] 1 and then enter an x value. Students should quickly see that as the cursor moves along the curve, the value of the function is the square root of the radicand of the function. Example 2 Graph y x 4. State the domain of the graph. Enter the equation in the Y= list. KEYSTROKES: 2nd [2 ] X,T,,n 4 ) GRAPH The value of the radicand will be positive when x 4 0, or when x 4. So the domain of x is {xx 4}. This graph looks like the graph of y x shifted left 4 units. [10, 10] scl: 1 by [10, 10] scl: 1 Assess Exercises 1–9. See pp. 639A–639B. Graph each equation and sketch the graph on your paper. State the domain of the graph. Then describe how the graph differs from the parent function y x . Ask: The graph of y x looks somewhat flat, as though it has a horizontal asymptote. Does it? Explain why or why not. As the value of x increases, the value of y, which is the square root of x, should continue to increase as well. There is no horizontal asymptote. 1. y x 1 4. y x5 7. y x 2. y x 3 5. y x 8. y 1x6 3. y x2 6. y 3x 9. y 2x 5 4 10. Is the graph of x y2 a function? Explain your reasoning. 10–11. See margin. 11. Does the equation x2 y2 1 determine y as a function of x? Explain. 12. Graph y x 1 x2 in the window defined by [2, 2] scl: 1 by [2, 2] scl: 1. Describe the graph. heart www.algebra1.com/other_calculator_keystrokes 604 Investigating Slope-Intercept Form 604 Chapter 11 Radical Expressions and Triangles Answers 10. No; you must consider the graph of y x and the graph of y x. This graph fails the vertical line test. For every value of x 0, there are two values for y. 11. No; the equation y 1 x 2 is not a function since there are both positive and negative values for y for each value of x. 604 Chapter 11 Radical Expressions and Triangles Lesson Notes The Pythagorean Theorem place chapter sphere art for each lesson at coordinates x= -2p y= –1p2 • Solve problems by using the Pythagorean Theorem. is the Pythagorean Theorem used in roller coaster design? Vocabulary • • • • hypotenuse legs Pythagorean triple corollary 1 Focus • Determine whether a triangle is a right triangle. The roller coaster Superman: Ride of Steel in Agawam, Massachusetts, is one of the world’s tallest roller coasters at 208 feet. It also boasts one of the world’s steepest drops, measured at 78 degrees, and it reaches a maximum speed of 77 miles per hour. You can use the Pythagorean Theorem to estimate the length of the first hill. 5-Minute Check Transparency 11-4 Use as a quiz or review of Lesson 11-3. 208 feet Mathematical Background notes are available for this lesson on p. 584D. 78˚ THE PYTHAGOREAN THEOREM In a right triangle, the side opposite the right angle is called the hypotenuse . This side is always the longest side of a right triangle. The other two sides are called the legs of the triangle. hypotenuse leg To find the length of any side of a right triangle when the lengths of the other two are known, you can use a formula developed by the Greek mathematician Pythagoras. leg The Pythagorean Theorem Study Tip • Words Triangles If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Sides of a triangle are represented by lowercase letters a, b, and c. c a is the Pythagorean Theorem used in roller coaster design? Ask students: • What shape is being used to estimate the length of the first hill? a right triangle • What information in the problem is most likely useful in finding the length of the hill? What information is extraneous? The height of the hill and the angle of the drop is probably useful, and the maximum speed is probably not useful. b • Symbols c2 a2 b2 Example 1 Find the Length of the Hypotenuse Find the length of the hypotenuse of a right triangle if a 8 and b 15. c2 a2 b2 c2 82 152 c2 289 Pythagorean Theorem a 8 and b 15 Simplify. c 289 Take the square root of each side. c 17 Disregard 17. Why? The length of the hypotenuse is 17 units. Lesson 11-4 The Pythagorean Theorem 605 Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 661–662 • Skills Practice, p. 663 • Practice, p. 664 • Reading to Learn Mathematics, p. 665 • Enrichment, p. 666 • Assessment, pp. 699, 701 Graphing Calculator and Spreadsheet Masters, p. 44 Parent and Student Study Guide Workbook, p. 86 School-to-Career Masters, p. 22 Transparencies 5-Minute Check Transparency 11-4 Real-World Transparency 11 Answer Key Transparencies Technology AlgePASS: Tutorial Plus, Lesson 32 Interactive Chalkboard Lesson x-x Lesson Title 605 Example 2 Find the Length of a Side 2 Teach Find the length of the missing side. In the triangle, c 25 and b 10 units. THE PYTHAGOREAN THEOREM In-Class Examples c2 a2 b2 252 Power Point® 102 Pythagorean Theorem b 10 and c 25 25 a 625 a2 100 Evaluate squares. 525 a2 1 Find the length of the hypote- 525 a nuse of a right triangle if a 18 and b 24. The length of the hypotenuse is 30 units. 22.91 a Subtract 100 from each side. Use a calculator to evaluate . 525 10 Use the positive value. To the nearest hundredth, the length of the leg is 22.91 units. 2 Find the length of the Whole numbers that satisfy the Pythagorean Theorem are called Pythagorean triples. Multiples of Pythagorean triples also satisfy the Pythagorean Theorem. Some common triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). missing side. b a2 Standardized Example 3 Pythagorean Triples Test Practice Multiple-Choice Test Item 16 B What is the area of triangle ABC? 9 A The length of the leg is approximately 13.23 units. C 96 units2 160 units2 B D 120 units2 196 units2 20 A 3 What is the area of triangle 12 C XYZ? C Read the Test Item Z 1 The area of a triangle is A bh. In a right triangle, the legs form the base and 2 height of the triangle. Use the measures of the hypotenuse and the base to find the height of the triangle. 35 X 28 Y A 94 units2 B 128 units2 C 294 units2 D 588 units2 Concept Check Solve the Test Item Step 1 Test-Taking Tip Memorize the common Pythagorean triples and check for multiples such as (6, 8, 10). This will save you time when evaluating square roots. 4 3 12 4 4 16 4 5 20 The height of the triangle is 16 units. Pythagorean Theorem Ask students to rewrite the formula for the Pythagorean Theorem in a2 b2 terms of c. c Check to see if the measurements of this triangle are a multiple of a common Pythagorean triple. The hypotenuse is 4 5 units, and the leg is 4 3 units. This triangle is a multiple of a (3, 4, 5) triangle. Step 2 Find the area of the triangle. 1 2 1 A 12 16 2 A bh A 96 Area of a triangle b 12 and h 16 Simplify. The area of the triangle is 96 square units. Choice A is correct. 606 Chapter 11 Radical Expressions and Triangles Example 3 Caution students that this is a two-step problem. First they must find the missing side length, then they must calculate the area of the triangle. Also stress to students that it is important to read each answer choice carefully before selecting the correct answer. In Example 3, choices B and C look very similar, and one could mistakenly choose D, 196, when the correct answer is A, 96. Standardized Test Practice 606 Chapter 11 Radical Expressions and Triangles RIGHT TRIANGLES A statement that can be easily proved using a theorem is often called a corollary. The following corollary, based on the Pythagorean Theorem, can be used to determine whether a triangle is a right triangle. RIGHT TRIANGLES In-Class Example Teaching Tip Point out to students that with Pythagorean triples, the greatest value is always the measure of the hypotenuse, and the two lesser values are the measures of the two legs. Corollary to the Pythagorean Theorem If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 a2 b2, then the triangle is a right triangle. If c2 a2 b2, then the triangle is not a right triangle. Example 4 Check for Right Triangles 4 Determine whether the following side measures form right triangles. Determine whether the following side measures form right triangles. a. 20, 21, 29 Since the measure of the longest side is 29, let c 29, a 20, and b 21. Then determine whether c2 a2 b2. c2 a2 b2 292 202 Pythagorean Theorem 212 841 841 1. leg b. 8, 10, 12 Since the measure of the longest side is 12, let c 12, a 8, and b 10. Then determine whether c2 a2 b2. Pythagorean Theorem 122 82 102 144 164 Concept Check GUIDED PRACTICE KEY Exercises Examples 4–9 10, 11 12 1, 2 4 3 Guided Practice Add. a2 b2, hypotenuse leg 2. Compare the lengths of the sides. The hypotenuse is the longest side, which is always the side opposite the right angle. a 8, b 10, and c 12 144 64 100 Multiply. Since b. 27, 36, 45 right triangle Answers Add. Since c2 a2 b2, the triangle is a right triangle. c2 a. 7, 12, 15 not a right triangle a 20, b 21, and c 29 841 400 441 Multiply. c2 a2 b2 Power Point® the triangle is not a right triangle. 1. OPEN ENDED Draw a right triangle and label each side and angle. Be sure to indicate the right angle. See margin. 2. Explain how you can determine which angle is the right angle of a right triangle if you are given the lengths of the three sides. See margin. 3. Write an equation you could use to find the length of the diagonal d of a square with side length s. d 2s 2 or d s 2 Find the length of each missing side. If necessary, round to the nearest hundredth. 18.44 4. c 12 5. 41 a 9 40 14 www.algebra1.com/extra_examples Lesson 11-4 The Pythagorean Theorem 607 Differentiated Instruction Kinesthetic Give students blocks, algebra tiles, or some other square or cube-shaped manipulatives. Have them use the manipulatives and what they know about the Pythagorean Theorem and Pythagorean triples to construct a right triangle, placing the manipulatives along the outside of the triangle. Have students construct the triangle on a piece of paper, then trace inside the blocks to transfer the triangle to the paper. Then ask them to use a protractor to confirm that the right angle is 90°. Lesson 11-4 The Pythagorean Theorem 607 If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 3 Practice/Apply 7. a 11, c 61, b ? 60 8. b 13, c 233 , a ? 8 9. a 7, b 4, c ? 8.06 65 Determine whether the following side measures form right triangles. Justify your answer. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 11. • include an explanation of the Pythagorean Theorem and examples of how to use it to find the sides or hypotenuse of a right triangle. • include any other item(s) that they find helpful in mastering the skills in this lesson. 6. a 10, b 24, c ? 26 10. 4, 6, 9 No; 42 62 92. Standardized Test Practice 11. 16, 30, 34 Yes; 162 302 342. 12. In right triangle XYZ, the length of Y Z is 6, and the length of the hypotenuse is 8. Find the area of the triangle. A 67 units2 A 30 units2 B C 40 units2 48 units2 D ★ indicates increased difficulty Practice and Apply Homework Help For Exercises See Examples 13–30 31–36 37–40 1, 2 4 3 Find the length of each missing side. If necessary, round to the nearest hundredth. 14.14 13. 14. 7 53 28 c a 15 11.40 15. 9 45 c Extra Practice See page 845. 5 16. About the Exercises … 175 5 Organization by Objective • The Pythagorean Theorem: 13–30, 37–40 • Right Triangles: 31–36 Odd/Even Assignments Exercises 13–36 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 44 requires the use of the Internet or another reference source. 13.08 17. 14 a 180 b 20 99 101 b If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 27. 253 15.91 28. 124 11.14 19. a 16, b 63, c ? 65 20. a 16, c 34, b ? 30 21. b 3, a 112 , c ? 11 22. a 15 , b 10 , c ? 5 10.72 115 b 77 , c 12, a ? 67 8.19 6.71 45 a 4, b 11 , c ? 27 5.20 23. c 14, a 9, b ? 24. a 6, b 3, c ? 25. 26. 27. a 225 , b 28 , c ? 28. a 31 , c 155 , b ? 29. a 8x, b 15x, c ? 17x 30. b 3x, c 7x, a ? 40x 6.32x Determine whether the following side measures form right triangles. Justify your answer. Assignment Guide 31. 30, 40, 50 Yes; 302 402 502. Basic: 13–37 odd, 41–43, 49–68 Average: 13–39 odd, 41–43, 45, 48–68 Advanced: 14–40 even, 44, 46–62 (optional: 63–68) 32. 6, 12, 18 No; 62 122 182. 34. 45, 60, 75 Yes; 452 602 752. 2 2 2 35. 15, 31, 16 Yes; 15 31 16 . 36. 4, 7, 65 Yes; 42 72 65 2. 33. 24, 30, 36 No; 242 302 362. Use an equation to solve each problem. If necessary, round to the nearest hundredth. 37. Find the length of a diagonal of a square if its area is 162 square feet. 18 ft 608 Chapter 11 Radical Expressions and Triangles Answer c 2 4 4 48. The area of the largest semicircle is c 2. 4 The sum of the other two areas is (a 2 b 2). Using the Pythagorean Theorem, c 2 a 2 b 2, we can show that the sum of the two small areas is equal to the area of the largest semicircle. 608 42.1318. Chapter 11 Radical Expressions and Triangles ★ 38. A right triangle has one leg that is 5 centimeters longer than the other leg. NAME ______________________________________________ DATE p. 661 (shown) and p. 662 The Pythagorean Theorem The Pythagorean Theorem The side opposite the right angle in a right triangle is called the hypotenuse. This side is always the longest side of a right triangle. The other two sides are called the legs of the triangle. To find the length of any side of a right triangle, given the lengths of the other two sides, you can use the Pythagorean Theorem. ★ 39. Find the length of the diagonal of the cube if each side of the cube is 4 inches long. 4 3 in. or about 6.93 in. ★ 40. The ratio of the length of the hypotenuse to the length B If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then c2 a2 b2. Pythagorean Theorem Example 1 Find the length of the hypotenuse of a right triangle if a 5 and b 12. 112.41 m c2 c2 c2 c c ROLLER COASTERS For Exercises 41–43, use the following information and the figure. Suppose a roller coaster climbs 208 feet higher than its starting point making a horizontal advance of 360 feet. When it comes down, it makes a horizontal advance of 44 feet. a2 b2 52 122 169 169 13 208 ft b A Example 2 Find the length of a leg of a right triangle if a 8 and c 10. c2 a2 b2 102 82 b2 100 64 b2 36 b2 b 36 b 6 Pythagorean Theorem a 5 and b 12 Simplify. Take the square root of each side. The length of the hypotenuse is 13. Pythagorean Theorem a 8 and c 10 Simplify. Subtract 64 from each side. Take the square root of each side. The length of the leg is 6. Exercises Find the length of each missing side. If necessary, round to the nearest hundredth. 1. 2. 3. 100 25 c a 110 25 c Lesson 11-4 30 360 ft 41. How far will it travel to get to the top of the ride? 415.8 ft 40 50 44 ft Roller Coasters 42. How far will it travel on the downhill track? 212.6 ft Roller Coaster Records in the U.S. Fastest: Millennium Force (2000), Sandusky, Ohio; 92 mph Tallest: Millenium Force (2000), Sandusky, Ohio; 310 feet Longest: California Screamin’ (2001), Anaheim, California; 6800 feet Steepest: Hypersonic XLC (2001), Doswell, Virginia; 90° 43. Compare the total horizontal advance, vertical height, and total track length. 45.83 4. a 10, b 12, c ? 44. RESEARCH Use the Internet or other reference to find the measurements of your favorite roller coaster or a roller coaster that is at an amusement park close to you. Draw a model of the first drop. Include the height of the hill, length of the vertical drop, and steepness of the hill. See students’ work. 5. a 9, b 12, c ? 15.62 6. a 12, b ?, c 16 15 10.58 8. a ?, b 8 , c 18 7. a ?, b 6, c 8 5.29 9. a 5 , b 10 , c ? 3.16 3.87 NAME ______________________________________________ DATE Skills Practice, 11-4 Practice (Average) ____________ PERIOD _____ p. 663 and Practice, p. 664 (shown) The Pythagorean Theorem Find the length of each missing side. If necessary, round to the nearest hundredth. 1. a 2. 3. c 32 45. SAILING A sailboat’s mast and boom form a right angle. The sail itself, called a mainsail, is in the shape of a right triangle. If the edge of the mainsail that is attached to the mast is 100 feet long and the edge of the mainsail that is attached to the boom is 60 feet long, what is the length of the longest edge of the mainsail? 116.6 ft 35.36 If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. The roller coaster makes a total horizontal advance of 404 feet, reaches a vertical height of 208 feet, and travels a total track length of 628.3 feet. 12 4 11 19 b 60 68 mast mainsail 15.49 11.31 If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 5. a 28, b 96, c ? 100 4. a 24, b 45, c ? 51 6. b 48, c 52, a ? 20 7. c 27, a 18, b ? 245 15.65 8. b 14, c 21, a ? boom 10. a 75 , b 6 , c ? 9 405 20.12 9. a 20 , b 10, c ? 120 10.95 11. b 9x, c 15x, a ? 12x Determine whether the following side measures form right triangles. Justify your answer. 12. 11, 18, 21 13. 21, 72, 75 No; 112 182 212. Yes; 212 722 752. 15. 9, 10, 161 14. 7, 8, 11 ROOFING For Exercises 46 and 47, refer to the figures below. No; 72 82 2 No; 92 102 (161 ) . 112. 16. 9, 210 , 11 5 ft ? ft c a C of the shorter leg in a right triangle is 8:5. The hypotenuse measures 144 meters. Find the length of the longer leg. Source: www.coasters2k.com ____________ PERIOD _____ Study Guide andIntervention Intervention, 11-4 Study Guide and The hypotenuse is 25 centimeters long. Find the length of each leg of the triangle. 15 cm, 20 cm 17. 7 , 22 , 15 2 2 Yes; 92 2(10 ) 112. 2 2 Yes; (7 ) (22 ) (15 ) . 18. STORAGE The shed in Stephan’s back yard has a door that measures 6 feet high and 3 feet wide. Stephan would like to store a square theater prop that is 7 feet on a side. Will it fit through the door diagonally? Explain. No; the greatest length that will fit through the door is 45 6.71 ft. 24 ft (entire span) SCREEN SIZES For Exercises 19–21, use the following information. The size of a television is measured by the length of the screen’s diagonal. 19. If a television screen measures 24 inches high and 18 inches wide, what size television is it? 30-in. television 46. Determine the missing length shown in the rafter. 13 ft 20. Darla told Tri that she has a 35-inch television. The height of the screen is 21 inches. What is its width? 28 in. 47. If the roof is 30 feet long and it hangs an additional 2 feet over the garage walls, how many square feet of shingles are needed for the entire garage roof? 900 ft2 21. Tri told Darla that he has a 5-inch handheld television and that the screen measures 2 inches by 3 inches. Is this a reasonable measure for the screen size? Explain. No; if the screen measures 2 in. by 3 in., then its diagonal is only about 3.61 in. NAME ______________________________________________ DATE ____________ PERIOD _____ Reading 11-4 Readingto to Learn Learn Mathematics Mathematics, p. 665 The Pythagorean Theorem ★ 48. CRITICAL THINKING Compare the area of the Pre-Activity largest semicircle to the areas of the two smaller semicircles. Justify your reasoning. See margin. ELL How is the Pythagorean Theorem used in roller coaster design? Read the introduction to Lesson 11-4 at the top of page 605 in your textbook. The diagram in the introduction shows a right triangle and part of the roller coaster. Which side of the right triangle has a length approximately equal to the length of the first hill of the roller coaster? c a the longest side, which is the side opposite the right angle b Reading the Lesson Complete each sentence. 1. The words leg and hypotenuse refer to the sides of a Lesson 11-4 The Pythagorean Theorem 609 triangle. 3. The longest side of a right triangle is called the hypotenuse leg of the right triangle. Write an equation that you could solve to find the missing side length of each right triangle. NAME ______________________________________________ DATE ____________ PERIOD _____ 4. 5. 6. 10 11-4 Enrichment Enrichment, 9 p. 666 10 6 9 10 c2 92 102 Pythagorean Triples Recall the Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a c b A a2 b2 c2 Note that c is the length of the hypotenuse. 9 16 25 25 25 The integers 3, 4, and 5 satisfy the Pythagorean theorem and can be the lengths of the sides of a right triangle. 32 42 52 Furthermore, for any positive integer n, the numbers 3n, 4n, and 5n satisfy the Pythagorean theorem. For n 2: 62 82 102 36 64 100 100 100 102 a2 92 7. Suppose you are given three positive numbers. Explain how you can decide whether these numbers are the lengths of the sides of a right triangle. B C 102 62 b2 Sample answer: Put the numbers in order from least to greatest. Square the three numbers. See whether the sum of the squares of the two smaller numbers is equal to the square of the largest number. If it is, then the numbers are the lengths of the sides of a right triangle. Helping You Remember 8. Think of a word or phrase that you can associate with the Pythagorean Theorem to help you remember the equation c2 a2 b2. Sample answer: Think of the word cab. The letters of this word are the same as those in the equation c 2 a 2 b 2. If three numbers satisfy the Pythagorean theorem, they are called a Lesson 11-4 The Pythagorean Theorem 609 Lesson 11-4 www.algebra1.com/self_check_quiz right 2. In a right triangle, each of the two sides that form the right angle is a of the right triangle. 49. CRITICAL THINKING A model of a part of a roller coaster is shown. Determine the total distance traveled from start to finish and the maximum height reached by the roller coaster. 1081.7 ft, 324.5 ft 4 Assess Open-Ended Assessment 80 ft Modeling Have students use pencils, uncooked spaghetti, or twigs to model right triangles. Ask students to explain the Pythagorean Theorem and relate it to their triangle. 180 ft 750 ft 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How is the Pythagorean Theorem used in roller coaster design? Include the following in your answer: • an explanation of how the height, speed, and steepness of a roller coaster are related, and • a description of any limitations you can think of in the design of a new roller coaster. Standardized Test Practice 51. Find the area of XYZ. C X A 65 units2 B 185 units2 C 45 units2 D 90 units2 15 52. Find the perimeter of a square whose diagonal measures 10 centimeters. B Assessment Options A C 102 cm B 202 cm 252 cm D 80 cm Z 2x x Y Maintain Your Skills Mixed Review Solve each equation. Check your solution. (Lesson 11-3) 53. y 12 144 54. 3s 126 1764 55. 4 2v 1 3 17 12 Simplify each expression. (Lesson 11-2) 56. 72 62 57. 7z 10z 3z 58. 821 37 21 7 Simplify. Assume that no denominator is equal to zero. (Lesson 8-2) 1 58 26a4b7c5 2a 2b 3 59. 3 55 or 3125 60. d7 7 61. 5 13a2b4c3 d c8 Answer 610 start 100 ft 130 ft 50 ft PREREQUISITE SKILL Students will learn about the Distance Formula in Lesson 11-5. In order to use the Distance Formula, students must be able to simplify radical expressions. Use Exercises 63–68 to determine your students’ familiarity with simplifying radical expressions. 50. Engineers can use the Pythagorean Theorem to find the total length of the track, which determines how much material and land area they need to build the attraction. Answers should include the following. • A tall hill requires more track length both going uphill and downhill, which will add to the total length of the tracks. Tall, steep hills will increase the speed of the roller coaster. So a coaster with a tall, steep first hill will have more speed and a longer track length. • The steepness of the hill and speed are limited for safety and to keep the cars on the track. 120 ft finish Getting Ready for Lesson 11-5 Quiz (Lessons 11-3 and 11-4) is available on p. 699 of the Chapter 11 Resource Masters. Mid-Chapter Test (Lessons 11-1 through 11-4) is available on p. 701 of the Chapter 11 Resource Masters. 381.2 ft 62. AVIATION Flying with the wind, a plane travels 300 miles in 40 minutes. Flying against the wind, it travels 300 miles in 45 minutes. Find the air speed of the plane. (Lesson 7-4) 425 mph Getting Ready for the Next Lesson PREREQUISITE SKILL Simplify each expression. (To review simplifying radical expressions, see Lesson 11-1.) 63. (6 3 )2 (8 4)2 5 65. (5 3 )2 (2 9)2 53 67. (4 5)2 (4 3)2 610 Chapter 11 Chapter 11 Radical Expressions and Triangles Radical Expressions and Triangles 64. (10 4)2 ( 13 5 )2 10 130 66. (9 5)2 (7 3 )2 253 68. (20 5)2 ( 2 6)2 17 Lesson Notes The Distance Formula • Find the distance between two points on the coordinate plane. 1 Focus • Find a point that is a given distance from a second point in a plane. can the distance between two points be determined? Study Tip Reading Math AC is the measure of AC and BC is the measure of BC. Consider two points A and B in the coordinate plane. Notice that a right triangle can be formed by drawing lines parallel to the axes through the points at A and B. These lines intersect at C forming a right angle. The hypotenuse of this triangle is the distance between A and B. You can A determine the length of the legs of this triangle and use the Pythagorean Theorem to find the distance between the two points. Notice that AC is the difference of the y-coordinates, and BC is the difference of the x-coordinates. 2 (BC )2. So, (AB)2 (AC)2 (BC)2, and AB (AC) y B O x C THE DISTANCE FORMULA You can find the distance between any two points in the coordinate plane using a similar process. The result is called the Distance Formula. The Distance Formula • Words • Model The distance d between any two points with coordinates (x1, y1) (x2 x1)2 (y2 y1)2. and (x2, y2) is given by d A (x 1, y 1) y B (x 2, y 2) O x 5-Minute Check Transparency 11-5 Use as a quiz or review of Lesson 11-4. Mathematical Background notes are available for this lesson on p. 584D. can the distance between two points be determined? Ask students: • What must you know in order to use the Pythagorean Theorem to determine the distance between the two points? You must know the lengths of the two legs in order to find the distance between the two points, which is the hypotenuse. • How do you find the length of the vertical leg? Find the difference in the y-coordinates of A and B. • How do you find the length of the horizontal leg? Find the difference in the x-coordinates of A and B. Example 1 Distance Between Two Points Find the distance between the points at (2, 3) and (4, 6). d (x2 x1)2 (y2 y1)2 Distance Formula (4 2)2 (6 3 )2 (x1, y1) (2, 3) and (x2, y2) (4, 6) Simplify. 45 Evaluate squares and simplify. (6)2 32 35 or about 6.71 units Lesson 11-5 The Distance Formula 611 Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 667–668 • Skills Practice, p. 669 • Practice, p. 670 • Reading to Learn Mathematics, p. 671 • Enrichment, p. 672 Parent and Student Study Guide Workbook, p. 87 Science and Mathematics Lab Manual, pp. 85–90 Transparencies 5-Minute Check Transparency 11-5 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 611 Example 2 Use the Distance Formula 2 Teach GOLF Tracy hits a golf ball that lands 20 feet short and 8 feet to the right of the cup. On her first putt, the ball lands 2 feet to the left and 3 feet beyond the cup. Assuming that the ball traveled in a straight line, how far did the ball travel on her first putt? THE DISTANCE FORMULA In-Class Examples Power Point® Draw a model of the situation on a coordinate grid. If the cup is at (0, 0), then the location of the ball after the first hit is (8, 20). The location of the ball after the first putt is (2, 3). Use the Distance Formula. Teaching Tip Point out to students that it does not matter which of the two points are designated (x1, y1) or (x2, y2). 2 (x2 x (y2 y1)2 d 1) (10)2 232 points at (1, 2) and (3, 0). The distance is 25, or about 4.47 units. Simplify. 629 or about 25 feet 2 BIATHLON Julianne is sight- 8642 O 5 10 15 20 25 y 2 4 6 8x (8, 20) FIND COORDINATES Suppose you know the coordinates of a point, one Golf There are four major tournaments that make up the “grand slam” of golf: Masters, U.S. Open, British Open, and PGA Championship. In 2000, Tiger Woods became the youngest player to win the four major events (called a career grand slam) at age 24. Source: PGA coordinate of another point, and the distance between the two points. You can use the Distance Formula to find the missing coordinate. Example 3 Find a Missing Coordinate Find the value of a if the distance between the points at (7, 5) and (a, 3) is 10 units. (x2 x1)2 (y2 y1)2 Distance Formula d 10 (a 7 )2 ( 3 5)2 Let x2 a, x1 7, y2 3, y1 5, 10 (a and d 10. 10 1 4a 4 9 64 Evaluate squares. 10 a2 1 4a 113 Simplify. 7)2 (8)2 a2 FIND COORDINATES In-Class Example 15 10 (2, 3) 5 (x , y ) (8, 20), (2 8)2 [3 ( 20)]2 (x1, y1) (2, 3) 2 2 1 Find the distance between the ing in her rifle for an upcoming biathlon competition. Her first shot is 2 inches to the right and 7 inches below the bull’s-eye. What is the distance between the bull’s-eye and where her first shot hit the target? The distance is 53 or about 7.28 inches. Distance Formula 102 a2 1 4a 113 Square each side. 100 a2 14a 113 Simplify. 2 Power Point® 0 3 Find the value of a if the distance between the points at (2, 1) and (a, 4) is 5 units. 2 or 6 a2 14a 13 Subtract 100 from each side. 0 (a 1)(a 13) a10 Factor. or a 13 0 a1 a 13 Zero Product Property The value of a is 1 or 13. Answers 1. The values that are subtracted are squared before being added and the square of a negative number is always positive. The sum of two positive numbers is positive, so the distance will never be negative. 2. See students’ graph; the distance from A to B equals the distance from B to A. Using the Distance Formula, the solution is the same no matter which ordered pair is used first. 3. See students’ diagrams; there are exactly two points that lie on the line y 3 that are 10 units from the point (7, 5). Concept Check 1–3. See margin. 1. Explain why the value calculated under the radical sign in the Distance Formula will never be negative. 2. OPEN ENDED Plot two ordered pairs and find the distance between their graphs. Does it matter which ordered pair is first when using the Distance Formula? Explain. 3. Explain why there are two values for a in Example 3. Draw a diagram to support your answer. 612 Chapter 11 Radical Expressions and Triangles Unlocking Misconceptions Not only does it not matter which point is designated (x1, y1) and (x2, y2) when using the Distance Formula, it also does not matter in which order the x- and y-coordinates are subtracted. For example, (x x1)2 (y2 y1)2 will produce the same results as 2 (x x2)2 (y1 y2)2 or (x x2)2 (y2 y1)2. Have students 1 1 test each method with two points. 612 Chapter 11 Radical Expressions and Triangles Guided Practice 4. (5, 1), (11, 7) 10 GUIDED PRACTICE KEY Exercises Examples 4–7 8, 9 10–12 1 3 2 Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 6. (2, 2), (5, 1) 32 4.24 5. (3, 7), (2, 5) 13 7. (3, 5), (6, 4) 3.16 10 Find the possible values of a if the points with the given coordinates are the indicated distance apart. 8. (3, 1), (a, 7); d 10 9 or 3 Applications 9. (10, a), (1, 6); d 145 2 or 14 10. GEOMETRY An isosceles triangle has two sides of equal length. Determine whether triangle ABC with vertices A(3, 4), B(5, 2), and C(1, 5) is an isosceles triangle. yes; AB 68 , BC 85 , AC 85 FOOTBALL For Exercises 11 and 12, use the information at the right. 11. 25.5 yd, 25 yd Quarterback 10 yd 5 yd Practice and Apply See Examples 13–26, 33, 34 27–32, 35, 36 37–42 1 13. (12, 3), (8, 3) 20 14. (0, 0), (5, 12) 13 3 15. (6, 8), (3, 4) 5 16. (4, 2), (4, 17) 17 2 17. (3, 8), (5, 4) 45 8.94 18. (9, 2), (3, 6) 213 7.21 Extra Practice See page 845. Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 19. (8, 4), (3, 8) 41 6.40 2 10 21. (4, 2), 6, 3.33 3 3 4 1 13 23. , 1, 2, or 1.30 5 2 10 25. 45, 7, 65, 1 214 7.48 Have students— • include examples of how to use the Distance Formula. • include any other item(s) that they find helpful in mastering the skills in this lesson. 10 20 30 40 ★ indicates increased difficulty For Exercises Study Notebook 25 yd 11. A quarterback can throw the football to one of the two receivers. Find the distance from the quarterback to each receiver. 12. What is the distance between the two receivers? 20.6 yd Homework Help 3 Practice/Apply 20. (2, 7), (10, 4) 185 13.60 17 1 22. 5, , (3, 4) or 4.25 4 4 3 2 74 24. 3, , 4, 1.23 7 7 7 26. 52, 8, 72, 10 23 3.46 Find the possible values of a if the points with the given coordinates are the indicated distance apart. 27. (4, 7), (a, 3); d 5 1 or 7 28. (4, a), (4, 2); d 17 17 or 13 29. (5, a), (6, 1); d 10 2 or 4 30. (a, 5), (7, 3); d 29 2 or 12 About the Exercises … Organization by Objective • The Distance Formula: 13–26, 33, 34, 37–42 • Find Coordinates: 27–32, 35, 36 Odd/Even Assignments Exercises 13–32 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 13–31 odd, 37–39, 43–68 Average: 13–37 odd, 38, 39, 43–68 Advanced: 14–36 even, 40–62 (optional: 63–68) 31. (6, 3), (3, a); d 130 10 or 4 32. (20, 5), (a, 9); d 340 2 or 38 ★ 33. Triangle ABC has vertices at A(7, 4), B(1, 2), and C(5, 6). Determine whether the triangle has three, two, or no sides that are equal in length. two; AB BC 10 34. 157 101 ; ★ 34. If the diagonals of a trapezoid have the same length, then the trapezoid is The trapezoid is not isosceles. Find the lengths of the diagonals of trapezoid ABCD with vertices isosceles. A(2, 2), B(10, 6), C(9, 8), and D(0, 5) to determine if it is isosceles. www.algebra1.com/extra_examples Lesson 11-5 The Distance Formula 613 Differentiated Instruction Interpersonal Place students in pairs. Have each person in a pair drop a penny on a coordinate grid, and record the coordinates of the point closest to where the center of the penny landed. Then have the students work together to find the distance between the two points, using the Distance Formula. Lesson 11-5 The Distance Formula 613 NAME ______________________________________________ DATE ★ 35. Triangle LMN has vertices at L(4, 3), M(2, 5), and N(13, 10). If the distance ____________ PERIOD _____ Study Guide andIntervention Intervention, 11-5 Study Guide and from point P(x, 2) to L equals the distance from P to M, what is the value of x? 3 p. 667 (shown) The Distance Formulaand p. 668 The Distance Formula The Pythagorean Theorem can be used to derive the Distance Formula shown below. The Distance Formula can then be used to find the distance between any two points in the coordinate plane. ★ 36. Plot the points Q(1, 7), R(3, 1), S(9, 3), and T(7, d). Find the value of d that makes each side of QRST have the same length. 9 The distance between any two points with coordinates (x1, y1) and (x2, y2) is given by Distance Formula (x2 x1)2 (y2 y1)2. d Example 1 Find the distance between the points at (5, 2) and (4, 5). d (x2 x1)2 ( y2 y1) 2 Distance Formula (4 ( 5))2 (5 2)2 (x1, y1) (5, 2), (x2, y2) (4, 5) 92 32 Simplify. 81 9 Evaluate squares and simplify. Example 2 Jill draws a line segment from point (1, 4) on her computer screen to point (98, 49). How long is the segment? 37. FREQUENT FLYERS To determine the mileage between cities for their frequent flyer programs, some airlines superimpose a coordinate grid over the United States. An ordered pair on the grid represents the location of each airport. The units of this grid are approximately equal to 0.316 mile. So, a distance of 3 units on the grid equals an actual distance of 3(0.316) or 0.948 mile. Suppose the locations of two airports are at (132, 428) and (254, 105). Find the actual distance between these airports to the nearest mile. 109 mi d (x2 x1)2 ( y2 y1) 2 (98 1)2 (49 4)2 972 452 90 The distance is 90 , or about 9.49 units. 9409 2025 11,43 4 The segment is about 106.93 units long. Exercises Find the distance between each pair of points whose coordinates are given. Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 3. (2, 8), (7, 3) 2. (0, 0), (6, 8) 25 ; 4.47 6. (3, 4), (4, 4) 5. (1, 5), (8, 4) 82 ; 9.06 17 7. (1, 4), (3, 2) 7 25 ; 4.47 34 ; 5.83 130 ; 11.40 12. (3, 4), (4, 16) 11. (3, 4), (0, 0) 173 ; 13.15 193 ; 13.89 5 13. (1, 1), (3, 2) 14. (2, 0), (3, 9) 15. (9, 0), (2, 5) 5 ; 2.24 82 ; 9.06 74 ; 8.60 17. (1, 3), (8, 21) 18. (3, 5), (1, 8) 16. (2, 7), (2, 2) 41 ; 6.40 38. Kelly has her first class in Rhodes Hall and her second class in Fulton Lab. How far does she have to walk between her first and second class? 0.53 mi 9. (2, 6), (7, 1) 8. (0, 0), (3, 5) 10. (2, 5), (0, 8) COLLEGE For Exercises 38 and 39, use the map of a college campus. 106 ; 10.30 10 4. (6, 7), (2, 8) 657 ; 25.63 Lesson 11-5 1. (1, 5), (3, 1) 5 NAME ______________________________________________ DATE Skills Practice, 11-5 Practice (Average) ____________ PERIOD _____ Find the distance between each pair of points whose coordinates are given. Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 3. (4, 6), (3, 9) 10 3.16 5. (0, 4), (3, 2) 35 6.71 1 2 6. (13, 9), (1, 5) 410 12.65 5 2 13 1 2 1 2 9. 2, , 1, 100 5 3 8. (1, 7), , 6 1.67 7. (6, 2), 4, or 2.50 2 7 2.65 1 20 M I NNE S OTA 40 Find the possible values of a if the points with the given coordinates are the indicated distance apart. 60 0 15. (6, 7), (a, 4); d 18 a 3 or 9 16. (4, 1), (a, 8); d 50 a 5 or 3 17. (8, 5), (a, 4); d 85 a 6 or 10 18. (9, 7), (a, 5); d 29 a 14 or 4 Duluth Three players are warming up for a baseball game. Player B stands 9 feet to the right and 18 feet in front of Player A. Player C stands 8 feet to the left and 13 feet in front of Player A. 16 20 41. Find the distance between the following pairs of cities: Minneapolis and St. Cloud, St. Paul and Rochester, Minneapolis and Eau Claire, and Duluth and St. Cloud. Eau Claire Rochester 42. A radio station in St. Paul has a broadcast range of 75 miles. Which cities shown on the map can receive the broadcast? B (9, 18) 80 12 C (8, 13) Fulton Lab 1 mi 8 Eau Claire, (71, 8); Rochester, (27, 58) St. Paul Minneapolis 60 y 3 mi 8 40. Estimate the coordinates for Duluth, St. Cloud, Eau Claire, and Rochester. Duluth, (44, 116); St. Cloud, (46, 39); St. Cloud 40 BASEBALL For Exercises 19–21, use the following information. Rhodes Hall GEOGRAPHY For Exercises 40–42, use the map at the left that shows part of Minnesota and Wisconsin. A coordinate grid has been superimposed on the map with the origin at St. Paul. The grid lines are 20 miles apart. Minneapolis is at (7, 3). 80 20 14. (2, 5), (a, 7); d 15 a 7 or 11 19. Draw a model of the situation on the coordinate grid. Assume that Player A is located at (0, 0). 40 60 3 12. (22 , 1), (32 , 3) 32 4.24 13. (4, 1), (a, 5); d 10 a 4 or 12 0 80 42 1.89 10. , 1 , 2, 3 3 2 1.41 11. (3 , 3), (23 , 5) 120 100 80 60 40 20 120 4. (3, 8), (7, 2) 229 10.77 Undergraduate Library minutes to walk between the two buildings. 170 13.04 2. (0, 9), (7, 2) 1 mi 4 39. She has 12 minutes between the end of her first class and the start of her second class. If she walks an average of 3 miles per hour, will she make it to her second class on time? Yes; it will take her 10.6 p. 669 and Practice, p.Formula 670 (shown) The Distance 1. (4, 7), (1, 3) 5 Campus Map all cities except Duluth 8 4 20. To the nearest tenth, what is the distance between Players A and B and between Players A and C? 20.1 ft; 15.3 ft A(0, 0) 4 O 4 8 x 8 21. What is the distance between Players B and C? 17.7 ft 22. MAPS Maria and Jackson live in adjacent neighborhoods. If they superimpose a coordinate grid on the map of their neighborhoods, Maria lives at (9, 1) and Jackson lives at (5, 4). If each unit on the grid is equal to approximately 0.132 mile, how far apart do Maria and Jackson live? about 1.96 mi NAME ______________________________________________ DATE ____________ PERIOD _____ Reading 11-5 Readingto to Learn Learn Mathematics ELL Mathematics, p. 671 The Distance Formula Pre-Activity How can the distance between two points be determined? 41. MinneapolisSt. Cloud, 53 mi; St. Paul-Rochester, 64 mi; MinneapolisRochester, 70 mi; Duluth-St. Cloud, 118 mi 43. CRITICAL THINKING Plot A(4, 4), B(7, 3), and C(4, 0), and connect them to form triangle ABC. Demonstrate two different ways to show whether ABC is a right triangle. See margin. 44. WRITING IN MATH How can the distance between two points be determined? Read the introduction to Lesson 11-5 at the top of page 611 in your textbook. What are the coordinates of points A, B, and C? Include the following in your answer: • an explanation how the Distance Formula is derived from the Pythagorean Theorem, and • an explanation why the Distance Formula is not needed to find the distance between points P(24, 18) and Q(24, 10). A (4, 3), B (3, 2), and C (4, 2) Reading the Lesson 1. Suppose you want to use the Distance Formula to find the distance between (6, 4) and (2, 1). Use (x1, y1) (6, 4) and (x2, y2) (2, 1). Complete the equations by writing the correct numbers in the blanks. a. x1 b. d 6 y1 4 x2 2 y2 y A 1 B O ( 2 6 )2 ( 1 4 )2 x 614 Chapter 11 Radical Expressions and Triangles 2. Suppose you want to use the Distance Formula to find the distance between (3, 7) and (9, 2). Use (x1, y1) (3, 7) and (x2, y2) (9, 2). Complete the equations by writing the correct numbers in the blanks. a. x1 b. d 3 y1 7 x2 9 Answer the question that was posed at the beginning of the lesson. See pp. 639A–639B. y2 2 ( 9 3 )2 ( 2 7 )2 3. A classmate is using the Distance Formula to find the distance between two points. She (2 5)2 (7 11)2. has done everything correctly so far, and her equation is d This equation will give her the distance between what two points? (5, 11) and (2, 7) Helping You Remember 4. Sometimes it is easier to remember a formula if you can state it in words. How can you state the Distance Formula in easy-to-remember words? Sample answer: Square the difference between the x-coordinates, then add that to the square of the difference between the y-coordinates. Take the square root of the sum to find the distance between the two points. NAME ______________________________________________ DATE 11-5 Enrichment Enrichment, ____________ PERIOD _____ p. 672 A Space-Saving Method Two arrangements for cookies on a 32 cm by 40 cm cookie sheet are shown at the right. The cookies have 8-cm diameters after they are baked. The centers of the cookies are on the vertices of squares in the top arrangement. In the other, the centers are on the vertices of equilateral triangles. Which arrangement is more economical? The triangle arrangement is more economical, because it contains one more cookie. 8 cm In the square arrangement, rows are placed every 8 cm. At what intervals are rows placed in the triangle arrangement? 20 cookies Look at the right triangle labeled a, b, and c. A leg a of the triangle is the radius of a cookie, or 4 cm. The hypotenuse c is the sum of two radii, or 8 cm. Use the Pythagorean theorem to find b, the interval of the rows. c2 a2 b2 82 42 b2 64 16 b2 614 Chapter 11 Radical Expressions and Triangles b? a c Standardized Test Practice 45. Find the distance between points at (6, 11) and (2, 4). B A 16 units B 17 units C 18 units D 19 units 4 Assess Open-Ended Assessment 46. Find the perimeter of a square ABCD if two of the vertices are A(3, 7) and B(3, 4). B A 12 units B 125 units C 95 units D 45 units Speaking Have students describe a real-world situation in which the Distance Formula could be used to find the distance between two points. If students have trouble thinking of situations, suggest that they think of anything that involves measurement such as construction, art, sports, and so on. Have them explain how the Distance Formula is used in the situation. Maintain Your Skills Mixed Review If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. (Lesson 11-4) 47. a 7, b 24, c ? 25 48. b 30, c 34, a ? 16 49. a 7, c 16 , b ? 3 50. a 13 , b 50 , c ? 37 7.94 Solve each equation. Check your solution. (Lesson 11-3) 51. p 2 8 p 11 52. r5r1 4 COST OF DEVELOPMENT For Exercises 54–56, use the graph that shows the amount of money being spent on worldwide construction. (Lesson 8-3) 54. Asia, 1,113,000,000,000; Europe, 1,016,000,000,000; U.S./Canada, 884,000,000,000; Latin America, 241,000,000,000; Middle East, 101,200,000,000; Africa, 56,100,000,000. 55. Asia, 1.113 1012; Europe, 1.016 1012; U.S./Canada, 8.84 1011; Latin America, 2.41 1011; Middle East, 1.012 1011; Africa, 5.61 1010. Getting Ready for the Next Lesson 54. Write the value shown for each continent or region listed in standard notation. 53. 5t2 29 2t 3 {2, 10} USA TODAY Snapshots® PREREQUISITE SKILL Students will learn about similar triangles in Lesson 11-6. In order to determine whether triangles are similar, they must be able to solve proportions. Use Exercises 63–68 to determine your students’ familiarity with solving proportions. Global spending on construction The worldwide construction industry, valued at $3.41 trillion in 2000, grew 5.8% since 1998. Breakdown by region: $1.113 trillion $1.016 trillion 55. Write the value shown for each continent or region in scientific notation. $884 billion 56. How much more money is being spent in Asia than in Latin America? $8.72 1011 $241 billion $101.2 $56.1 billion billion or $872 billion Asia Europe U.S./ Latin Middle Canada America East Africa Source: Engineering News Record By Shannon Reilly and Frank Pompa, USA TODAY Solve each inequality. Then check your solution and graph it on a number line. 57–62. See pp. 639A–639B for graphs. 57. 8 m 1 {mm 9} 58. 3 10 k {kk 7} 59. 3x 2x 3 {xx 3} 60. v (4) 6 {vv 2} (Lesson 6-1) 61. r 5.2 3.9 {rr 9.1} 1 1 2 62. s ss 6 3 2 PREREQUISITE SKILL Solve each proportion. (To review proportions, see Lesson 3-6.) 20 5 64. 8 x 3 63. 6 4 2 6 8 65. 12 9 x x2 3 67. 7 7 1 www.algebra1.com/self_check_quiz Getting Ready for Lesson 11-6 Answer 43. Compare the slopes of the two potential legs to determine whether the slopes are negative reciprocals of each other. You can also compute the lengths of the three sides and determine whether the square of the longest side length is equal to the sum of the squares of the other two side lengths. Neither test holds true in this case because the triangle is not a right triangle. x 2 x 10 66. 15 18 12 6 2 68. 5 x4 3 Lesson 11-5 The Distance Formula 615 Online Lesson Plans USA TODAY Education’s Online site offers resources and interactive features connected to each day’s newspaper. Experience TODAY, USA TODAY’s daily lesson plan, is available on the site and delivered daily to subscribers. This plan provides instruction for integrating USA TODAY graphics and key editorial features into your mathematics classroom. Log on to www.education.usatoday.com. Lesson 11-5 The Distance Formula 615 Lesson Notes 1 Focus 5-Minute Check Transparency 11-6 Use as a quiz or review of Lesson 11-5. Similar Triangles • Determine whether two triangles are similar. • Find the unknown measures of sides of two similar triangles. Vocabulary • similar triangles Mathematical Background notes are available for this lesson on p. 584D. Building on Prior Knowledge In Lesson 3-6, students learned how to solve proportions. In this lesson, students will use their knowledge of proportions to determine whether two triangles are similar. are similar triangles related to photography? Ask students: • If you photograph two people, one of whom is twice the height of the other, what are the heights of their images in the photograph? One image will be twice the height of the other. • How would you describe this relationship using ratios? The ratio of the heights of the people is the same as the ratio of the heights of the images. • What do you call an equation containing two ratios? a proportion are similar triangles related to photography? When you take a picture, the image of the object being photographed is projected by the camera lens onto the film. The height of the image on the film can be related to the height of the object using similar triangles. Film Lens Distance from lens to object Distance from lens to film SIMILAR TRIANGLES Similar triangles have the same shape, but not necessarily the same size. There are two main tests for similarity. • If the angles of one triangle and the corresponding angles of a second triangle have equal measures, then the triangles are similar. • If the measures of the sides of two triangles form equal ratios, or are proportional, then the triangles are similar. Study Tip The triangles below are similar. This is written as ABC DEF. The vertices of similar triangles are written in order to show the corresponding parts. Reading Math E The symbol is read is similar to. B 2 cm A 4 cm 83˚ 2.5 cm 56˚ 41˚ C 3 cm D corresponding angles 83˚ 5 cm 56˚ 41˚ F 6 cm corresponding sides 2 1 AB A B and D E → A and D 4 2 DE BC 2.5 1 B C and E F → EF 5 2 AC 3 1 A C and D F → DF 6 2 B and E C and F Similar Triangles Study Tip Reading Math Arcs are used to show angles that have equal measures. • Words If two triangles are similar, then the measures of their corresponding sides are proportional, and the measures of their corresponding angles are equal. • Symbols If ABC DEF, • Model AB BC AC then . DE EF DF B E C A D 616 Chapter 11 Radical Expressions and Triangles Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 673–674 • Skills Practice, p. 675 • Practice, p. 676 • Reading to Learn Mathematics, p. 677 • Enrichment, p. 678 • Assessment, p. 700 Parent and Student Study Guide Workbook, p. 88 Prerequisite Skills Workbook, pp. 61–62 Transparencies 5-Minute Check Transparency 11-6 Answer Key Transparencies Technology Interactive Chalkboard F Example 1 Determine Whether Two Triangles Are Similar 2 Teach Determine whether the pair of triangles is similar. Justify your answer. Remember that the sum of the measures of the angles in a triangle is 180°. N SIMILAR TRIANGLES The measure of P is 180° (51° 51°) or 78°. M In MNO, N and O have the same measure. Let x the measure of N and O. x x 78° 180° 2x 102° x 51° In-Class Example 78˚ P Teaching Tip Tell students to look closely at all the angles and sides of the two triangles for clues about whether the triangles are similar. In Example 1, the same angles are not marked on both triangles, but there is sufficient information to determine that the angles are congruent. O 51˚ R Power Point® 51˚ Q So N 51° and O 51°. Since the corresponding angles have equal measures, MNO PQR. 1 Determine whether the pair FIND UNKNOWN MEASURES Proportions can be used to find the of triangles is similar. Justify your answer. measures of the sides of similar triangles when some of the measurements are known. Y 15 Example 2 Find Missing Measures Study Tip Corresponding Vertices Always use the corresponding order of the vertices to write proportions for similar triangles. Find the missing measures if each pair of triangles below is similar. X a. Since the corresponding angles have equal measures, TUV WXY. The lengths of the corresponding sides are proportional. 10 XY Corresponding sides of similar WX UV triangles are proportional. TU a 16 WX a, XY 16, TU 3, UV 4 3 4 4a 48 a 12 4b 96 b 24 U 3 98˚ 47˚ 6 T Find the cross products. 4 98˚ a 47˚ W 51˚ b Y Divide each side by 4. FIND UNKNOWN MEASURES a. E B 10 A E 6m xm G D y 18 77 26 77 9m I D 18 27 77 The missing measure is 15. www.algebra1.com/extra_examples C 8 C BE 10, CD x, AE 6, AD 9 Divide each side by 6. Power Point® each pair of triangles below is similar. Corresponding sides of similar triangles are proportional. 15 x C 16 2 Find the missing measures if b. ABE ACD Find the cross products. 8 In-Class Example Find the cross products. 90 6x B 16 The missing measures are 12 and 24. BE AE CD AD 10 6 x 9 Z 24 The corresponding sides of the triangles are proportional, so the triangles are similar. V X Divide each side by 4. XY Corresponding sides of similar WY UV triangles are proportional. TV b 16 WY b, XY 16, TV 6, UV 4 6 4 A 12 26 77 H x The missing measures are 27 and 12. Lesson 11-6 Similar Triangles 617 b. Z 10 Y 4 X cm cm 3 cm W a cm V The missing measure is 7.5. Lesson 11-6 Similar Triangles 617 In-Class Example Example 3 Use Similar Triangles to Solve a Problem Power Point® SHADOWS Jenelle is standing near the Washington Monument in Washington, D.C. The shadow of the monument is 302.5 feet, and Jenelle’s shadow is 3 feet. If Jenelle is 5.5 feet tall, how tall is the monument? 3 SHADOWS Richard is standing next to the General Sherman Giant Sequoia tree in Sequoia National Park. The shadow of the tree is 22.5 meters, and Richard’s shadow is 53.6 centimeters. If Richard’s height is 2 meters, how tall is the tree? The tree is about 84 m tall. 3 Practice/Apply Note: Not drawn to scale x ft The shadows form similar triangles. Write a proportion that compares the heights of the objects and the lengths of their shadows. 5.5 ft Washington Monument Let x the height of the monument. Jenelle’s shadow → monument’s shadow → The monument has a shape of an Egyptian obelisk. A pyramid made of solid aluminum caps the top of the monument. 3 5.5 302.5 x 3 ft 302.5 ft ← Jenelle’s height ← monument’s height 3x 1663.75 Cross products x 554.6 feet Divide each side by 3. The height of the monument is about 554.6 feet. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 11. • include examples of how to determine whether two triangles are similar, and how to find unknown measures using triangles. • include any other item(s) that they find helpful in mastering the skills in this lesson. Concept Check 1. If the measures of the angles of one triangle equal the measures of the corresponding angles of another triangle, and the lengths of the sides are proportional, then the two triangles are similar. 2. OPEN ENDED Draw a pair of similar triangles. List the corresponding angles and the corresponding sides. See pp. 639A–639B. 3. FIND THE ERROR Russell and Consuela are comparing the similar triangles below to determine their corresponding parts. See margin. Russell ΩX = ΩT ΩY = ΩU ΩZ = ΩV πXYZ , πTUV Y Consuela }X = }V }Y = }U }Z = }T X Z U πXYZ , πVUT T V Who is correct? Explain your reasoning. Guided Practice GUIDED PRACTICE KEY Exercises Examples 4, 5 6–9 10 1 2 3 FIND THE ERROR Remind students to pay close attention to the arcs that denote which angles are congruent. Determine whether each pair of triangles is similar. Justify your answer. 4. No; the angle measures are not equal. 84˚ 46˚ 5. 35˚ 55˚ Yes; the angle measures are equal. 46˚ For each set of measures given, find the measures of the missing sides if ABC DEF. 6. c 15, d 7, e 9, f 5 a 21, b 27 7. a 18, c 9, e 10, f 6 b 15, d 12 25 30 8. a 5, d 7, f 6, e 5 b , c 7 7 9. a 17, b 15, c 10, f 6 d 10.2, A Answers 3. Consuela; the arcs indicate which angles correspond. The vertices of the triangles are written in order to show the corresponding parts. 1. Explain how to determine whether two triangles are similar. 618 Chapter 11 Radical Expressions and Triangles e9 B c E a f b C D Differentiated Instruction Naturalist Have students use the method in Example 3 to find the heights of trees that are native to your area. Make sure students record the location and type of tree along with the height. Students will need tape measures and will need to take measurements on a sunny day. 618 Chapter 11 Radical Expressions and Triangles d e F Application 10. SHADOWS If a 25-foot flagpole casts a shadow that is 10 feet long and the nearby school building casts a shadow that is 26 feet long, how high is the building? 65 ft Organization by Objective • Similar Triangles: 11–16 • Find Unknown Measures: 17–32 ★ indicates increased difficulty Practice and Apply Homework Help For Exercises See Examples 11–16 17–24 25–32 1 2 3 Determine whether each pair of triangles is similar. Justify your answer. 11. yes 21˚ 12. 54˚ no 21˚ 105˚ 13. no 42˚ 60˚ 50˚ Extra Practice See page 845. 14. yes 15. 44˚ no 50˚ 16. 88˚ yes 55˚ 55˚ x 45˚ 56˚ Odd/Even Assignments Exercises 11–24 are structured so that students practice the same concepts whether they are assigned odd or even problems. 60˚ 50˚ 60˚ 79˚ 45˚ 45˚ Assignment Guide 2x 80˚ Basic: 11–27 odd, 33–61 Average: 11–27 odd, 29, 30, 33–61 Advanced: 12–28 even, 29–55 (optional: 56–61) All: Practice Quiz 2 (1–10) 11–16. See margin for justifications. For each set of measures given, find the measures of the missing sides if KLM NOP. 17. k 9, n 6, o 8, p 4 ᐉ 12, m 6 L 18. k 24, ᐉ 30, m 15, n 16 o 20, p 10 55 22 19. m 11, p 6, n 5, o 4 k , ᐉ 6 112 3 84 20. k 16, ᐉ 13, m 12, o 7 n , p 13 13 21. n 6, p 2.5, ᐉ 4, m 1.25 k 3, o 8 m k K Answers M l O 22. p 5, k 10.5, ᐉ 15, m 7.5 n 7, o 10 p 23. n 2.1, ᐉ 4.5, p 3.2, o 3.4 k 2.8, m 4.2 N 24. m 5, k 12.6, o 8.1, p 2.5 ᐉ 16.2, n 6.3 About the Exercises … n P o 25. Determine whether the following statement is sometimes, always, or never true. If the measures of the sides of a triangle are multiplied by 3, then the measures of the angles of the enlarged triangle will have the same measures as the angles of the original triangle. always 11. The angle measures are equal. 12. The angle measures are not equal. 13. The angle measures are not equal. 14. The angle measures are equal. 15. The angle measures are not equal. 16. The angle measures are equal. 26. PHOTOGRAPHY Refer to the diagram of a camera at the beginning of the lesson. Suppose the image of a man who is 2 meters tall is 1.5 centimeters tall on film. If the film is 3 centimeters from the lens of the camera, how far is the man from the camera? 4 m 1 3 27. 3 in. 27. BRIDGES Truss bridges use triangles in their support beams. Mark plans to make a model of a truss bridge in the scale 1 inch 12 feet. If the height of the triangles on the actual bridge is 40 feet, what will the height be on the model? 28. BILLIARDS Lenno is playing billiards on a table like the one shown at the right. He wants to strike the cue ball at D, bank it at C, and hit another ball at the mouth of pocket A. Use similar triangles to find where Lenno’s cue ball should strike the rail. 24 in. from pocket B A 84 in. B x 42 in. D 10 in. 28 in. C E F Lesson 11-6 Similar Triangles 619 Lesson 11-6 Similar Triangles 619 NAME ______________________________________________ DATE CRAFTS For Exercises 29 and 30, use the following information. Melinda is working on a quilt pattern containing isosceles triangles whose sides measure 2 inches, 2 inches, and 2.5 inches. ____________ PERIOD _____ Study Guide andIntervention Intervention, 11-6 Study Guide and Similar Triangles RST is similar to XYZ. The angles of the two triangles have equal measure. They are called corresponding angles. The sides opposite the corresponding angles are called corresponding sides. Z X 30 S 60 60 Y 30 R T E If two triangles are similar, then the measures of their corresponding sides are proportional and the measures of their corresponding angles are equal. Similar Triangles ABC DEF B Lesson 11-6 p. 673 and p. 674 Similar(shown) Triangles ★ 29. She has several square pieces of material that measure 4 inches on each side. From each square piece, how many triangles with the required dimensions can she cut? 8 D F AB BC AC DE EF DF ★ 30. She wants to enlarge the pattern to make similar triangles for the center of A C Example 1 S Determine whether the pair of triangles is similar. Justify your answer. F 90 X G 45 R 75 50 45 T the quilt. What is the largest similar triangle she can cut from the square material? 4 by 4 by 5 in. Example 2 Determine whether the pair of triangles is similar. Justify your answer. H I 45 45 E 89 Y MIRRORS For Exercises 31 and 32, use the diagram and the following information. Viho wanted to measure the height of a nearby building. He placed a mirror on the pavement at point P, 80 feet from the base of the building. He then backed away until he saw an image of the top of the building in the mirror. J Z The measure of G 180° (90° 45°) 45°. The measure of I 180° (45° 45°) 90°. Since corresponding angles have equal measures, EFG HIJ. Since corresponding angles do not have the equal measures, the triangles are not similar. Exercises Determine whether each pair of triangles is similar. Justify your answer. 1. 2. 120 3. 90 30 45 30 30 60 60 Yes; corresponding angles have equal measures. No; corresponding angles do not have equal measures. 4. Yes; corresponding angles have equal measures. 5. 40 6. 45 30 Yes; corresponding angles have equal measures. ★ 31. If Viho is 6 feet tall and he is standing 9 feet from the mirror, how tall is the building? about 53 ft 120 110 30 20 80 45 30 115 55 No; corresponding angles do not have equal measures. No; corresponding angles do not have equal measures. NAME ______________________________________________ DATE ____________ PERIOD _____ Skills Practice, p. 675 and 11-6 Practice (Average) Practice, p. 676 (shown) Similar Triangles Determine whether each pair of triangles is similar. Justify your answer. 1. 2. U D P R Q 56 47 59 S G 80 C 31 T 47 F E H Yes; Q T 90°; P 180° (90° 31°) 59° S; U 180° (90° 59°) 31° R. Since the corresponding angles have equal measures, PQR STU. No; C 180° (47° 80°) 53°. Since FGH has a 56° angle, but CDE does not, corresponding angles do not all have equal measures, and the triangles are not similar. For each set of measures given, find the measures of the missing sides if ABC DEF. E f 3. c 4, d 12, e 16, f 8 a 6, b 8 D B d c F e A a b C 4. e 20, a 24, b 30, c 15 d 16, f 10 ★ 32. What assumptions did you make in solving the problem? 32. Viho’s eyes are 6 feet off the ground, Viho and the building CRITICAL THINKING For Exercises 33–35, use the following information. each create right The radius of one circle is twice the radius of another. 34–35. See margin. angles with the 33. Are the circles similar? Explain your reasoning. ground, and the two angles with 34. What is the ratio of their circumferences? Explain your reasoning. the ground at P have 35. What is the ratio of their areas? Explain your reasoning. equal measure. 33. Yes; all circles are similar because they 36. WRITING IN MATH Answer the question that was posed at the beginning of have the same shape. the lesson. See margin. How are similar triangles related to photography? 5. a 10, b 12, c 6, d 4 e 4.8, f 2.4 Include the following in your answer: • an explanation of the effect of moving a camera with a zoom lens closer to the object being photographed, and • a description of what you could do to fit the entire image of a large object on the picture. 8 3 6. a 4, d 6, e 4, f 3 c 2, b 15 2 7. b 15, d 16, e 20, f 10 a 12, c 32 3 44 3 8. a 16, b 22, c 12, f 8 d , e 5 2 33 14 11 2 35 6 9. a , b 3, f , e 7 c , d 10. c 4, d 6, e 5.625, f 12 a 2, b 1.875 11. SHADOWS Suppose you are standing near a building and you want to know its height. The building casts a 66-foot shadow. You cast a 3-foot shadow. If you are 5 feet 6 inches tall, how tall is the building? 121 ft Standardized Test Practice 12. MODELS Truss bridges use triangles in their support beams. Molly made a model of a truss bridge in the scale of 1 inch 8 feet. If the height of the triangles on the model is 4.5 inches, what is the height of the triangles on the actual bridge? 36 ft NAME ______________________________________________ DATE For Exercises 37 and 38, use the figure at the right. 37. Which statement is true? D ____________ PERIOD _____ ABC ADC ABC ACD ABC CAD none of the above A Reading 11-6 Readingto to Learn Learn MathematicsELL Mathematics, p. 677 Similar Triangles Pre-Activity B C How are similar triangles related to photography? Read the introduction to Lesson 11-6 at the top of page 616 in your textbook. D How would you describe the shapes and sizes of the figures in the diagram? Sample answer: The figures have the same shape, but they do not have the same size. Reading the Lesson 1. In similar triangles, the angles of the two triangles can be matched so that angles have equal measures triangle, then the two angles are not similar C . 3. If two triangles have the same size and shape, then the measures of the corresponding proportional AB DC AC BC A . 2. If the angles of one triangle do not have the same measures as the angles of a second sides are 620 Chapter 11 Radical Expressions and Triangles . Determine whether each pair of triangles is similar. Explain how you know that your answer is correct. Not similar; the angles cannot be matched so that corresponding angles have the same measures. 4. 58 30 NAME ______________________________________________ DATE 11-6 Enrichment Enrichment, 32 ____________ PERIOD _____ p. 678 60 Not similar; the sides cannot be matched to have corresponding sides proportional. 5. 7.5 6 3 8 6. Similar; the angles can be matched so that corresponding angles have equal measures. 125 35 35 125 A Curious Construction Many mathematicians have been interested in ways to construct the number . Here is one such geometric construction. 3.5 4 20 A 20 Helping You Remember 7. How can you use the idea that the corresponding sides of similar triangles are proportional to help you remember how to find the unknown lengths of the sides of similar triangles? Sample answer: Since the corresponding sides of similar C 7 – 8 In the drawing, triangles ABC and ADE are right triangles. The length of A D equals the length of AC and F B is parallel to EG . G gives a decimal approximation The length of B of the fractional part of to six decimal places. . Follow the steps to find the length of BG 1. Use the length of B C and the Pythagorean Theorem to find the length of AC . AC triangles are proportional, use a proportion to find the lengths of the unknown sides. 1 1.3287682 2 7 2 8 2. Find the length of A D . AD 620 Chapter 11 Radical Expressions and Triangles AC 1 3287682 F E 1 – 2 A 1 B G D C D 38. Which statement is always true? A Complete each sentence. corresponding B B D CB AD AC AB Maintain Your Skills Mixed Review Find the distance between each pair of points whose coordinates are given. Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. (Lesson 11-5) 40. (6, 3), (12, 5) 10 39. (1, 8), (2, 4) 5 41. (4, 7), (3, 12) 26 5.1 42. 1, 56 , 6, 76 7 Determine whether the following side measures form right triangles. Justify your answer. (Lesson 11-4) 43. 25, 60, 65 Yes; 252 602 652. 45. 49, 168, 175 Yes; 492 1682 1752. 44. 20, 25, 35 No; 202 252 352. 46. 7, 9, 12 No; 72 92 122. Arrange the terms of each polynomial so that the powers of the variable are in descending order. (Lesson 8-4) 48. 5x3 2x2 4x 7 47. 1 3x2 7x 3x2 7x 1 48. 7 4x 2x2 5x3 49. 6x 3 3x2 3x2 6x 3 50. abx2 bcx 34 x7 x7 abx2 bcx 34 Use elimination to solve each system of equations. (Lesson 7-3) 51. 2x y 4 (3, 2) xy5 52. 3x 2y 13 (5, 1) 2x 5y 5 53. 0.6m 0.2n 0.9 (1.5, 0) 0.3m 0.45 0.1n 1 1 54. x y 8 (6, 12) 3 2 1 1 x y 0 2 4 55. AVIATION An airplane passing over Sacramento at an elevation of 37,000 feet begins its descent to land at Reno, 140 miles away. If the elevation of Reno is 4500 feet, what should be the approximate slope of descent? (Hint: 1 mi 5280 ft) (Lesson 5-1) 232 ft/mi or about 0.044 Getting Ready for the Next Lesson PREREQUISITE SKILL Evaluate if a 6, b 5, and c 1.5. (To review evaluating expressions, see Lesson 1-2.) a 56. 4 c ac 9 59. b 5 or 1.8 5 b 57. or 0.83 a 6 10 b 60. or 1.1 ac 9 P ractice Quiz 2 2 ab 58. or 0.6 c 3 1 c 61. or 0.3 ac 3 Lessons 11-4 through 11-6 If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. (Lesson 11-4) 1. a 14, b 48, c ? 50 2. a 40, c 41, b ? 9 3. b 8, c 84 , a ? 25 4.47 4. a 5, b 8, c ? 4 Assess Open-Ended Assessment Modeling Construct a triangle using unsharpened pencils, uncooked spaghetti or other items that have a uniform length. Have students construct a triangle similar to yours and explain how they would show that the two triangles are similar. Getting Ready for Lesson 11-7 PREREQUISITE SKILL Students will learn about trigonometric ratios in Lesson 11-7. In order to compute trigonometric ratios, students must be able to divide rational numbers. Use Exercises 56–61 to determine your students’ familiarity with dividing rational numbers. Assessment Options Practice Quiz 2 The quiz provides students with a brief review of the concepts and skills in Lessons 11-4 through 11-6. Lesson numbers are given to the right of exercises or instruction lines so students can review concepts not yet mastered. Quiz (Lessons 11-5 and 11-6) is available on p. 700 of the Chapter 11 Resource Masters. 3.61 13 Find the distance between each pair of points whose coordinates are given. (Lesson 11-5) 17.49 306 22 2.83 5. (6, 12), (3, 3) 6. (1, 3), (5, 11) 10 7. (2, 5), (4, 7) 8. (2, 9), (5, 4) 178 13.34 Find the measures of the missing sides if BCA EFD. (Lesson 11-6) 9. b 10, d 7, e 2, f 3 a 35, c 15 10. a 12, c 9, d 8, e 12 b 18, f 6 www.algebra1.com/self_check_quiz Lesson 11-6 Similar Triangles 621 Answers 34. 2:1; Let the first circle have radius r and the larger have radius 2r. The circumference of the first is 2r and the other has circumference 2(2r ) 4r. 35. 4:1; The area of the first is r 2 and the area of the other is (2r)2 4r 2. 36. The size of an object on the film of a camera can be related to its actual size using similar triangles. Answers should include the following. • Moving the lens closer to the object (and farther from the film) makes the object appear larger. • Taking a picture of a building; you would need to be a great distance away to fit the entire building in the picture. Lesson 11-6 Similar Triangles 621 Algebra Activity A Preview of Lesson 11-7 A Preview of Lesson 11-7 Investigating Trigonometric Ratios Getting Started You can use paper triangles to investigate trigonometric ratios. Objective Construct right triangles with legs whose lengths are equal to a specific ratio, in order to investigate the relationship between the sides and the angles as a preview of trigonometric ratios. Collect the Data Materials ruler grid paper protractor Teach • Remind students that the triangles they are drawing are similar because the ratio of the corresponding sides is the same for each triangle. • Point out to students that because they drew the triangles by hand, the measures of the angles may not be exact, according to the protractor. However, the measures should be close to 35°, 55°, and 90°. Use a ruler and grid paper to draw several right triangles whose legs are in a 7:10 ratio. Include a right triangle with legs 3.5 units and 5 units, a right triangle with legs 7 units and 10 units, another with legs 14 units and 20 units, and several more right triangles similar to these three. Label the vertices of each triangle as A, B, and C, where C is at the right angle, B is opposite the longest leg, and A is opposite the shortest leg. Step 2 Copy the table below. Complete the first three columns by measuring the hypotenuse (side AB) in each right triangle you created and recording its length. Step 3 Calculate and record the ratios in the middle two columns. Round to the nearest tenth, if necessary. Step 4 Use a protractor to carefully measure angles A and B in each right triangle. Record the angle measures in the table. Side Lengths side BC side AC side AB Ratios BC:AC Angle Measures BC:AB angle A angle B angle C 0.6 0.6 0.6 0.6 0.6 0.6 35° 35° 35° 35° 35° 35° 55° 55° 55° 55° 55° 55° 90° 6.1 7:10 7 10 12.2 7:10 14 20 24.4 7:10 28 40 48.8 7:10 35 50 61 7:10 42 60 73.2 7:10 Sample answers are given in table. 3.5 5 90° 90° 90° 90° 90° Analyze the Data Assess Ask students to conjecture why knowing the relationship between the ratios of the sides of a triangle to each other and the angles might be useful. Sample answer: Knowing these relationships might allow you to calculate a missing side length or angle measure. Step 1 1. Examine the measures and ratios in the table. What do you notice? Write a sentence or two to describe any patterns you see. All ratios and angle measures are the same for any 7:10 right triangle. Make a Conjecture 2. For any right triangle similar to the ones you have drawn here, what will be the value of the ratio of the length of the shortest leg to the length of the longest leg? 7:10 3. If you draw a right triangle and calculate the ratio of the length of the shortest leg to the length of the hypotenuse to be approximately 0.573, what will be the measure of the larger acute angle in the right triangle? 55° 622 Investigating Slope-Intercept Form 622 Chapter 11 Radical Expressions and Triangles Resource Manager Teaching Algebra with Manipulatives Glencoe Mathematics Classroom Manipulative Kit • p. 1 (master for grid paper) • p. 23 (master for protractors) • p. 24 (master for rulers) • p. 191 (student recording sheet) • coordinate grid stamp • protractors • rulers 622 Chapter 11 Radical Expressions and Triangles Lesson Notes Trigonometric Ratios • Define the sine, cosine, and tangent ratios. 1 Focus • Use trigonometric ratios to solve right triangles. Vocabulary • • • • • • • trigonometric ratios sine cosine tangent solve a triangle angle of elevation angle of depression are trigonometric ratios used in surveying? 5-Minute Check Transparency 11-7 Use as a quiz or review of Lesson 11-6. Surveyors use triangle ratios called trigonometric ratios to determine distances that cannot be measured directly. Mathematical Background notes are available for this lesson on p. 584D. • In 1852, British surveyors measured the altitude of the peak of Mt. Everest at 29,002 feet using these trigonometric ratios. • In 1954, the official height became 29,028 feet, which was also calculated using surveying techniques. • On November 11, 1999, a team using advanced technology and the Global Positioning System (GPS) satellite measured the mountain at 29,035 feet. TRIGONOMETRIC RATIOS Trigonometry is an area of mathematics that involves angles and triangles. If enough information is known about a right triangle, certain ratios can be used to find the measures of the remaining parts of the triangle. Trigonometric ratios are ratios of the measures of two sides of a right triangle. Three common trigonometric ratios are called sine, cosine, and tangent. Trigonometric Ratios • Words measure of leg opposite A measure of hypotenuse sine of A measure of leg adjacent to A cosine of A Study Tip measure of hypotenuse Reading Math Notice that sine, cosine, and tangent are abbreviated sin, cos, and tan respectively. • Symbols measure of leg opposite A tangent of A measure of leg adjacent to A BC sin A AB AC cos A AB BC tan A AC B • Model leg opposite hypotenuse A A leg adjacent A are trigonometric ratios used in surveying? Ask students: • In the previous lesson, you calculated the height of the Washington Monument using ratios. What is required in order to make this calculation? Two similar triangles formed by the shadow of the monument and the shadow of a person standing next to it. You know the length of both shadows and the height of the person. You can use ratios to calculate the height of the monument. • Based on the Algebra Activity on the previous page, how do you suppose a surveyor could calculate the height of a mountain? Sample answer: A surveyor could use the angle between him or herself and the top of the mountain, and the distance to the mountain to construct a similar right triangle. Then based on the side measures of that similar triangle, the height of the mountain could be calculated using ratios. C Lesson 11-7 Trigonometric Ratios 623 Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 679–680 • Skills Practice, p. 681 • Practice, p. 682 • Reading to Learn Mathematics, p. 683 • Enrichment, p. 684 • Assessment, p. 700 Parent and Student Study Guide Workbook, p. 89 Teaching Algebra With Manipulatives Masters, pp. 23, 24, 192 Transparencies 5-Minute Check Transparency 11-7 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 623 Example 1 Sine, Cosine, and Tangent 2 Teach Find the sine, cosine, and tangent of each acute angle of RST. Round to the nearest ten thousandth. TRIGONOMETRIC RATIOS The trigonometric ratios are a concept that students will use in other classes such as Geometry, Algebra II, and Trigonometry. Any time students are introduced to such important concepts, have them record the concepts in their Study Notebooks for future reference. Additionally, the act of recording the concepts will help the students remember them. 18 17 Write each ratio and substitute the measures. Use a calculator to find each value. S 35 T New In-Class Example R opposite leg hypotenuse adjacent leg hypotenuse opposite leg adjacent leg cos R sin R 35 or 0.3287 18 opposite leg hypotenuse tan R adjacent leg hypotenuse 17 opposite leg adjacent leg cos T sin T 35 or 0.3480 17 18 or 0.9444 tan T 35 or 0.3287 17 18 or 0.9444 17 or 2.8735 35 18 You can use a calculator to find the values of trigonometric functions or to find the measure of an angle. On a graphing calculator, press the trigometric function key, and then enter the value. On a nongraphing scientific calculator, enter the value, and then press the function key. In either case, be sure your calculator is in degree mode. Consider cos 50°. Graphing Calculator COS 50 ENTER .6427876097 Power Point® Nongraphing Scientific Calculator KEYSTROKES: 50 COS .642787609 KEYSTROKES: Teaching Tip While it is true that a fraction with radicals in the denominator is not in simplest form, it is not necessary to rationalize the denominator if students are going to use a calculator to simplify the expression. Suggest that students use a cal10 3 Example 2 Find the Sine of an Angle Find sin 35° to the nearest ten thousandth. KEYSTROKES: SIN 35 ENTER .5735764364 Rounded to the nearest ten thousandth, sin 35° 0.5736. 30 3 culator to find and . Example 3 Find the Measure of an Angle Both produce the same result, and rationalization was not necessary. Find the measure of J to the nearest degree. Since the lengths of the opposite and adjacent sides are known, use the tangent ratio. 1 Find the sine, cosine, and opposite leg adjacent leg 6 9 tangent of each acute angle of DEF. Round to the nearest ten-thousandth. 9 KL 6 and JK 9 L Now use the TAN on a calculator to find the 6 measure of the angle whose tangent ratio is . –1 9 19 KEYSTROKES: 2nd [TAN1] 6 ⫼ 9 ENTER 33.69006753 To the nearest degree, the measure of J is 34°. E F 83 sin D 0.7293; cos D 0.6842; tan D 1.0659; sin F 0.6842; cos F 0.7293; tan F 0.9382 624 Chapter 11 Radical Expressions and Triangles Differentiated Instruction Auditory/Musical Divide the class into small groups. Challenge each group to devise a rap verse, limerick, poem, or a song that they can use to remember the trigonometric ratios. 624 Chapter 11 Radical Expressions and Triangles K 6 tan J Definition of tangent D 13 J SOLVE TRIANGLES You can find the missing measures of a right triangle if you know the measure of two sides of a triangle or the measure of one side and one acute angle. Finding all of the measures of the sides and the angles in a right triangle is called solving the triangle . Example 4 Solve a Triangle Find all of the missing measures in ABC. You need to find the measures of B, A C , and B C . A Step 1 Find the measure of B. The sum of the measures of the angles in a triangle is 180. 38˚ y in. 180° 90° 38° 52° Study Tip Verifying Right Triangles You can use the Pythagorean Theorem to verify that the sides are sides of a right triangle. C x in. B If students have trouble understanding the concept of the inverse of a trigono6 9 B C is about 7.4 inches long. Step 3 Find the value of y, which is the measure of the side adjacent to A. Use the cosine ratio. 9.5 y Teaching Tip metric function, write 4J on Evaluate sin 38°. Multiply by 12. y 12 y 0.7880 12 To make sure that their calculators are in degree mode, tell students to press MODE and then look at the third row to see whether Radian or Degree is highlighted. If Radian is highlighted, students should use the cursor to select Degree, then press ENTER . 2 Find cos 65° to the nearest sin 38° Definition of sine cos 38° Teaching Tip ten thousandth. 0.4226 Step 2 Find the value of x, which is the measure of the side opposite A. Use the sine ratio. 7.4 x Power Point® 12 in. The measure of B is 52°. x 12 x 0.6157 12 In-Class Examples Definition of cosine the chalkboard. Ask students to explain how you would solve this problem. They will say to divide both sides by 4. Point out that dividing by 4 is the inverse operation of multiplying J by 4. Therefore, to solve for J in the 6 9 Evaluate cos 38°. equation tan J , you perform Multiply by 12. the inverse of the tangent operation on both sides of the equation. A C is about 9.5 inches long. So, the missing measures are 52°, 7.4 in., and 9.5 in. 3 Find the measure of B to Trigonometric ratios are often used to find distances or lengths that cannot be measured directly. In these situations, you will sometimes use an angle of elevation or an angle of depression. An angle of elevation is formed by a horizontal line of sight and a line of sight above it. An angle of depression is formed by a horizontal line of sight and a line of sight below it. the nearest degree. 12 A B 20 C To the nearest degree, the measure of B is 53°. line of sight angle of depression angle of elevation SOLVE TRIANGLES In-Class Example www.algebra1.com/extra_examples Power Point® 4 Find all of the missing Lesson 11-7 Trigonometric Ratios 625 measures in DEF. D 15 cm x cm E 62 y cm F The missing measures are mF 28°, x 8 cm, and y 17 cm. Lesson 11-7 Trigonometric Ratios 625 Power Point® Make a Hypsometer Teaching Tip Problems involving angles of depression are solved in the same manner as problems with angles of elevation. It is just a different orientation of the angle. 110 170 120 130 140 150 160 • Tie one end of a piece of string to the middle of a straw. Tie the other end of string to a paper clip. • Tape a protractor to the side of the straw. Make Straw sure that the string hangs freely to create a vertical or plumb line. • Find an object outside that is too tall to measure 90 directly, such as a basketball hoop, a flagpole, or the school building. String • Look through the straw to the top of the object you are measuring. Find the angle measure where Paper Clip the string and protractor intersect. Determine the angle of elevation by subtracting this measurement from 90°. • Measure the distance from your eye level to the ground and from your foot to the base of the object you are measuring. 10 5 INDIRECT MEASUREMENT In the diagram, Barone is flying his model airplane 400 feet above him. An angle of depression is formed by a horizontal line of sight and a line of sight below it. Find the angles of depression at points A and B. 400 ft 10 0 In-Class Example 20 30 40 50 60 70 80 Analyze 1. See students’ work. 1. Make a sketch of your measurements. Use the equation 130 ft B A height of object x tan (angle of elevation) , where x represents distance distance of object from the ground to your eye level, to find the height of the object. 400 ft The angle of depression at point A is 45° and the angle of depression at point B is 37°. 2. The angle measured by the hypsometer is not the angle of elevation. It is the other acute angle formed in the triangle. So, to find the measure of the angle of elevation, subtract the reading on the hypsometer from 90 since the sum of the two acute angles in a right triangle is 90. 2. Why do you have to subtract the angle measurement on the hypsometer from 90° to find the angle of elevation? 3. Compare your answer with someone who measured the same object. Did your heights agree? Why or why not? See students’ work. Example 5 Angle of Elevation INDIRECT MEASUREMENT At point A, Umeko measured the angle of elevation to point P to be 27 degrees. At another point B, which was 600 meters closer to the cliff, Umeko measured the angle of elevation to point P to be 31.5 degrees. Determine the height of the cliff. Explore Draw a diagram to model the situation. Two right triangles, BPC and APC, are formed. You know the angle of elevation for each triangle. To determine the height of the cliff, find the length of P C , which is shared by both triangles. y 27˚ B A 31.5˚ 600 m xm Let y represent the distance from the top of the cliff P to its base C. Let x represent BC in the first triangle and let x 600 represent AC. Solve Write two equations involving the tangent ratio. y x and x tan 31.5° y y 600 x tan 27° (600 x)tan 27° y Chapter 11 Radical Expressions and Triangles Algebra Activity Materials string, drinking straw, paper clip, protractor, tape, meter sticks or tape measure Remind students that the angle of elevation is being measured from their eye level, not the ground. They must account for this distance when calculating the height of the object that they are measuring with their hypsometer. 626 Chapter 11 Radical Expressions and Triangles C Plan tan 31.5° 626 P Since both expressions are equal to y, use substitution to solve for x. x tan 31.5° (600 x) tan 27° Substitute. 3 Practice/Apply x tan 31.5° 600 tan 27° x tan 27° Distributive Property x tan 31.5° x tan 27° 600 tan 27° Subtract. x(tan 31.5° tan 27°) 600 tan 27° Isolate x. 600 tan 27° x tan 31.5° tan 27° Divide. x 2960 feet Use a calculator. Use this value for x and the equation x tan 31.5° y to solve for y. x tan 31.5° y Original equation 2960 tan 31.5° y Replace x with 2960. 1814 y Use a calculator. The height of the cliff is about 1814 feet. Examine Examine the solution by finding the angles of elevation. y x 1814 tan B 2960 y 600 x 1814 tan A 600 2960 tan A tan B B 31.5° A 27° Study Notebook Have students— • complete the definitions/examples for the remaining terms on their Vocabulary Builder worksheets for Chapter 11. • include examples of how to derive the trigonometric ratios from a right triangle, and how to use the ratios to solve triangles. • include any other item(s) that they find helpful in mastering the skills in this lesson. The solution checks. Answers Concept Check 2. See margin. 1. Explain how to determine which trigonometric ratio to use when solving for an unknown measure of a right triangle. See margin. 2. OPEN ENDED Draw a right triangle and label the measure of the hypotenuse and the measure of one acute angle. Then solve for the remaining measures. 3. Compare the measure of the angle of elevation and the measure of the angle of depression for two objects. What is the relationship between their measures? They are equal. Guided Practice GUIDED PRACTICE KEY Exercises Examples 4, 5 6–8 9–14 15–17 18 1 2 3 4 5 For each triangle, find sin Y, cos Y, and tan Y to the nearest ten thousandth. 4. Y 27 X sin Y 0.8, cos Y 0.6, tan Y 1.3333 36 45 5. W 10 26 sin Y 0.3846, cos Y 0.9231, tan Y 0.4167 1. If you know the measure of the hypotenuse, use sine or cosine, depending on whether you know the measure of the adjacent side or the opposite side. If you know the measures of the two sides, use tangent. 2. Sample answer: A 180° (90° 50°) or 40° AC 10 AC 7.66 X 24 Y BC 6.43 A Z 10 Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 6. sin 60° 0.8660 BC 10 sin 50° cos 50° 7. cos 75° 0.2588 8. tan 10° 0.1763 C 50˚ B Use a calculator to find the measure of each angle to the nearest degree. 9. sin W 0.9848 80° 10. cos X 0.6157 52° 11. tan C 0.3249 18° Lesson 11-7 Trigonometric Ratios 627 Lesson 11-7 Trigonometric Ratios 627 For each triangle, find the measure of the indicated angle to the nearest degree. About the Exercises … 12. Organization by Objective • Trigonometric Ratios: 19–51 • Solve Triangles: 52–65 19. sin R 0.6, cos R 0.8, tan R 0.75 20. sin R 0.3243, cos R 0.9459, tan R 0.3429 21. sin R 0.7241, cos R 0.6897, tan R 1.05 22. sin R 0.4706, cos R 0.8824, tan R 0.5333 23. sin R 0.5369, cos R 0.8437, tan R 0.6364 24. sin R 0.8182, cos R 0.5750, tan R 1.4230 9.7 7 ? 6 17° 22° 9.3 Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 15. A 16. C 17. B 18 m 42 in. 30˚ C A B A 60°, AC 21 in., BC 36.4 in. Application B B 35°, BC 5.7 in., AB 7.0 in. 35˚ Assignment Guide Answers 14. ? ? Odd/Even Assignments Exercises 19–60 are structured so that students practice the same concepts whether they are assigned odd or even problems. Basic: 19–49 odd, 53–59 odd, 61, 62, 66–79 Average: 19–59 odd, 61–64, 66–79 Advanced: 20–60 even, 63–79 15 13. 33° 13 A 55°, AC 12.6 m, A AB 22 m 18. DRIVING The percent grade of a road is the ratio of how much the road rises or falls in a given horizontal distance. If a road has a vertical rise of 40 feet for every 1000 feet horizontal distance, calculate the percent grade of the road and the angle of elevation the road makes with the horizontal. 4% grade, 2.3° 55˚ C 4 in. Trucks check brakes 6% grade ★ indicates increased difficulty Practice and Apply Homework Help For Exercises See Examples 19–24 25–33 34–51 52–60 61–65 1 2 3 4 5 19–24. See margin. For each triangle, find sin R, cos R, and tan R to the nearest ten thousandth. 19. 20. R T 10 F 35 8 S Extra Practice See page 846. 29 20 G R R 12 37 6 21. M 22. R 30 34 P 23. 24. K N 21 R 410 16 7 170 O 22 T 18 H 11 R U Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 25. sin 30° 0.5 26. sin 80° 0.9848 27. cos 45° 0.7071 28. cos 48° 0.6691 29. tan 32° 0.6249 30. tan 15° 0.2679 31. tan 67° 2.3559 32. sin 53° 0.7986 33. cos 12° 0.9781 Use a calculator to find the measure of each angle to the nearest degree. 628 628 Chapter 11 Radical Expressions and Triangles 34. cos V 0.5000 60° 35. cos Q 0.7658 40° 36. sin K 0.9781 78° 37. sin A 0.8827 62° 38. tan S 1.2401 51° 39. tan H 0.6473 33° 40. sin V 0.3832 23° 41. cos M 0.9793 12° 42. tan L 3.6541 75° Chapter 11 Radical Expressions and Triangles For each triangle, find the measure of the indicated angle to the nearest degree. 44. 45. 14 2 16 10 For each acute angle of a right triangle, certain ratios of side lengths are useful. These ratios are called trigonometric ratios. Three common ratios are the sine, cosine, and tangent, as defined at the right. 16 51° 47. 21 Trigonometric Ratios ? 39° 46. p. 679 (shown) Trigonometric Ratios and p. 680 10 ? 8° ? 57° 219 ? sin R 0.436 20 18 20 cos R 0.9 219 49. 15 ★ 50. 9 45 16 ? tan R 0.484 24 30 ? 56° 1. 45˚ C 8 ft A B 69°, AB 13.9 cm, BC 5 cm 20 in. B 13 cm 2. A 13 5 A 9 cm C A: 0.4455, 0.8911, 0.5000 B: 0.8911, 0.4455, 2.0000 A 56. A: 0.9231, 0.3846, 2.4000 B: 0.3846, 0.9231, 0.4167 Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 4. cos 47° 0.6820 5. tan 48° 1.1106 6. sin A 0.7547 49° 7. tan C 2.3456 67° 8. cos B 0.6947 46° 9. sin A 0.6589 41° 10. tan C 1.9832 63° 11. cos B 0.0136 89° NAME ______________________________________________ DATE p. 681 and Practice, p. Ratios 682 (shown) Trigonometric 2. P Y 3. 25 24 C In emergency situations, modern submarines are built to allow for a rapid surfacing, a technique called an emergency main ballast blow. 58. A 53°, B 37°, AB 10 ft B 50°, AC 12.3 ft, BC 10.3 ft B A A C B 38˚ C Source: www.howstuffworks.com T sin T 0.8944, cos T 0.4472, tan T 2 Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 24 in. B 52°, AC 30.7 in., AB 39 in. 4. sin 85° 0.9962 5. cos 5° 0.9962 6. tan 32.5° 0.6371 Use a calculator to find the measure of each angle to the nearest degree. 7. sin D 0.5000 30° 8. cos Q 0.1123 84° 9. tan B 4.7465 78° For each triangle, find the measure of the indicated angle to the nearest degree. B 10. 55° 13 11. 72° 12. ? 42 A 30°, B 60°, AB 5.2 cm 60. A 6 cm C 12 ft A 23°, B 67°, AB 13 ft 41° 12 9 B B 5 8 T sin T 0.96, cos T 0.28, tan T 3.4286 ? 59. A 8 7 R sin T 0.3243, cos T 0.9459, tan T 0.3429 16 C T 24 74 C 5 ft 8 ft 6 ft 9m X 40 16 ? Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 13. C 14. A B 51 15. B 64 C 13 cm 15 f 26 yd 30 ft C A A 39°, AC 20.2 yd, BC 16.4 yd A B B 26°, AC 5.7 cm, BC 11.7 cm A 60°, B 30°, BC 26.0 ft HIKING For Exercises 16 and 17, use the following information. The 10-mile Lower West Rim trail in Mt. Zion National Park ascends 2640 feet. C 3 cm 16. What is the average angle of elevation of the hike from the canyon floor to the top of the canyon? (Hint: Draw a diagram in which the length of the trail forms the hypotenuse of a right triangle. Convert feet to miles.) about 2.9° B 17. What is the horizontal distance covered on the hike? about 10 mi SUBMARINES For Exercises 61 and 62, use the following information. A submarine is traveling parallel to the surface of the water 626 meters below the surface. The sub begins a constant ascent to the surface so that it will emerge on the ? surface after traveling 4420 meters from the point of its initial ascent. 18. RAMP DESIGN An engineer designed the entrance to a museum to include a wheelchair ramp that is 15 feet long and forms a 6° angle with a sidewalk in front of the museum. How high does the ramp rise? about 1.6 ft NAME ______________________________________________ DATE Mathematics, p. 683 Trigonometric Ratios 626 m Pre-Activity ELL How are trigonometric ratios used in surveying? Read the introduction to Lesson 11-7 at the top of page 623 in your textbook. In which years were the measurements of the height of Mt. Everest in closest agreement? 4420 m 1954 and 1999 61. What angle of ascent did the submarine make? 8.1° Reading the Lesson 62. What horizontal distance did the submarine travel during its ascent? 4375 m 1. Complete each sentence. N MR a. The legs of triangle MNR are segments b. The hypotenuse of triangle MNR is www.algebra1.com/self_check_quiz ____________ PERIOD _____ Reading 11-7 Readingto to Learn Learn Mathematics Lesson 11-7 Trigonometric Ratios 629 MN and NR M . c. The leg opposite N is MR , and the leg adjacent to N is NR d. The leg opposite M is NR , and the leg adjacent to M is MR 2. Write K or S to complete each equation. NAME ______________________________________________ DATE 11-7 Enrichment Enrichment, ____________ PERIOD _____ p. 684 Modern Art The painting below, aptly titled, “Right Triangles,” was painted by that well-known artist Two-loose La-Rectangle. Using the information below, find the dimensions of this masterpiece. (Hint: The triangle that includes F is isosceles.) R . . S a. tan S KT TS b. sin S c. sin K ST KS d. cos S e. cos K KT KS f. tan K KT KS ST KS T ST KT 3. If you look straight in front of you and then up, the two lines of sight form an angle of elevation (depression/elevation). If you look straight in front of you and then down, the two lines of sight form an angle of 6 ft . depression (depression/elevation). E B Helping You Remember 4. What is one way you can remember the trigonometric ratios sine, cosine, and tangent? Sample answer: A well-known mnemonic device is SOH-CAH-TOA (sin opposite/hypotenuse, cos adjacent/hypotenuse, tan opposite/adjacent). A C Lesson 11-7 Trigonometric Ratios 629 K Lesson 11-7 Submarines AC 3.3 m, AB 9.6 m 20˚ 40˚ ____________ PERIOD _____ Skills Practice, 11-7 Practice (Average) 1. A 70°, 57. A B B 12 For each triangle, find sin T, cos T, and tan T to the nearest ten thousandth. C 16 ft T From part a above 70 55. B 2 19 Use a calculator to find the measure of each angle to the nearest degree. 21˚ A 63°, AC 9.1 in., BC 17.8 in. S r 219 , t 18 3. sin 45° 0.7071 54. 27˚ A 20 18 10.1 cm C Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. B R r 219 , s 20 B 4.5 cm 23 53. C A 45°, AB 11.3 ft, AC 8 ft a 90 C b Find the sine, cosine, and tangent of each acute angle. Round your answers to the nearest ten thousandth. 7° 52. A A b Exercises 25 37° tan A a c c Use TAN –1 on a calculator to find the measure of the angle whose tangent ratio is 0.484. KEYSTROKES: 2nd [TAN –1] .484 ENTER 25.82698212 or about 26° ★ 51. ? tangent of A B c b. Find the measure of R to the nearest degree. 32° 17 cos A b t 18, s 20 tan R 0.484 18 ? 36° sin A a cosine of A a. Find the sine, cosine, and tangent of R of RST. Round to the nearest thousandth. 8 5 ? sine of A Example 48. 21 25 ____________ PERIOD _____ Lesson 11-7 43. NAME ______________________________________________ DATE Study Guide andIntervention Intervention, 11-7 Study Guide and AVIATION For Exercises 63 and 64, use the following information. Germaine pilots a small plane on weekends. During a recent flight, he determined that he was flying 3000 feet parallel to the ground and that the ground distance to the start of the landing strip was 8000 feet. 4 Assess Open-Ended Assessment 63. What is Germaine’s angle of depression to the start of the landing strip? 20.6° Writing Ask students to sketch a right triangle, label the sides and angles with their measures, and find the trigonometric ratios for the two acute angles. 64. What is the distance between the plane in the air and the landing strip on the ground? 8544 ft Assessment Options Quiz (Lesson 11-7) is available on p. 700 of the Chapter 11 Resource Masters. ★ 65. FARMING Leonard and Alecia are building a new feed storage system on their farm. The feed conveyor must be able to reach a range of heights. It has a length of 8 meters, and its angle of elevation can be adjusted from 20° to 5°. Under these conditions, what range of heights is possible for an opening in the building through which feed can pass? 2.74 m to 0.7 m 66. See margin. 66. CRITICAL THINKING An important trigonometric identity is sin2 A cos2 A 1. Use the sine and cosine ratios and the Pythagorean Theorem to prove this identity. 67. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are trigonometric ratios used in surveying? Include the following in your answer: • an explanation of how trigonometric ratios are used to measure the height of a mountain, and • any additional information you need to know about the point from which you are measuring in order to find the altitude of a mountain. Answers a c b c 66. Let sin A and let cos A , where a and b are legs of a right triangle and c is the hypotenuse. Then sin2 A cos2 A Standardized Test Practice a2 b2 a2 b2 2 . Since the 2 c c c2 26 B 23 C 43 D 22 45° D 60° 4 69. What is the measure of Q? D c2 c becomes 2 or 1. Thus 630 R 68. RT is equal to TS. What is RS? A A Pythagorean Theorem states that a 2 b 2 c2, the expression sin2 A cos2 A 1. 67. If you know the distance between two points and the angles from these two points to a third point, you can determine the distance to the third point by forming a triangle and using trigonometric ratios. Answers should include the following. • If you measure your distance from the mountain and the angle of elevation to the peak of the mountain from two different points, you can write an equation using trigonometric ratios to determine its height, similar to Example 5. • You need to know the altitude of the two points you are measuring. For Exercises 68 and 69, use the figure at the right. A 25° B 30° C 2 Q T S Maintain Your Skills Mixed Review For each set of measures given, find the measures of the missing sides if KLM ∼ NOP. (Lesson 11-6) N 70. k 5, ᐉ 3, m 6, n 10 o 6, p 12 71. ᐉ 9, m 3, n 12, p 4.5 k 8, o 13.5 p K o m ᐉ M k L P n O Find the possible values of a if the points with the given coordinates are the indicated distance apart. (Lesson 11-5) 72. (9, 28), (a, 8); d 39 6 or 24 73. (3, a), (10, 1); d 65 5 or 3 Find each product. (Lesson 8-6) 74. c2(c2 3c) c4 3c3 75. s(4s2 9s 12) 4s3 9s2 12s 76. xy2(2x2 5xy 7y2) 2x3y2 5x2y3 7xy4 Use substitution to solve each system of equations. (Lesson 7-2) 77. a 3b 2 (11, 3) 4a 7b 23 630 Chapter 11 Radical Expressions and Triangles Chapter 11 Radical Expressions and Triangles 78. p q 10 (3, 7) 3p 2q 5 79. 3r 6s 0 (2, 1) 4r 10s 2 Reading Mathematics The Language of Mathematics Getting Started The language of mathematics is a specific one, but it borrows from everyday language, scientific language, and world languages. To find a word’s correct meaning, you will need to be aware of some confusing aspects of language. Confusing Aspect Have students think of words that have more than one meaning. As students respond, make a list of the words on the chalkboard along with their multiple definitions. Words Some words are used in English and in mathematics, but have distinct meanings. factor, leg, prime, power, rationalize Some words are used in English and in mathematics, but the mathematical meaning is more precise. difference, even, similar, slope Some words are used in science and in mathematics, but the meanings are different. divide, radical, solution, variable Some words are only used in mathematics. decimal, hypotenuse, integer, quotient Some words have more than one mathematical meaning. base, degree, range, round, square Sometimes several words come from the same root word. polygon and polynomial, radical and radicand Some mathematical words sound like English words. cosine and cosign, sine and sign, sum and some Some words are often abbreviated, but you must use the whole word when you read them. cos for cosine, sin for sine, tan for tangent Teach • Place students in small groups. Assign each group several of the words listed in this activity to define. Have students use their textbooks, dictionaries, or the Internet to find the definitions. Then, as you discuss each group of words, ask the groups to give the definitions. Assess Words in boldface are in this chapter. Study Notebook Reading to Learn 1–3. See margin. Ask students to summarize what they have learned about the language of mathematics. 1. How do the mathematical meanings of the following words compare to the everyday meanings? a. factor b. leg c. rationalize 2. State two mathematical definitions for each word. Give an example for each definition. a. degree b. range c. round 3. Each word below is shown with its root word and the root word’s meaning. Find ELL English Language Learners may benefit from writing key concepts from this activity in their Study Notebooks in their native language and then in English. three additional words that come from the same root. a. domain, from the root word domus, which means house b. radical, from the root word radix, which means root c. similar, from the root word similis, which means like Reading Mathematics The Language of Mathematics 631 Reading Mathematics The Language of Mathematics 631 Change art for numerals as necessary should come in at x=p6, y=1p6 reduce to 65% Study Guide and Review Vocabulary and Concept Check Vocabulary and Concept Check angle of depression (p. 625) angle of elevation (p. 625) conjugate (p. 589) corollary (p. 607) cosine (p. 623) Distance Formula (p. 611) extraneous solution (p. 599) • This alphabetical list of vocabulary terms in Chapter 11 includes a page reference where each term was introduced. • Assessment A vocabulary test/review for Chapter 11 is available on p. 698 of the Chapter 11 Resource Masters. hypotenuse (p. 605) leg (p. 605) Pythagorean triple (p. 606) radical equation (p. 598) radical expression (p. 586) radicand (p. 586) State whether each sentence is true or false. If false, replace the underlined word, number, expression, or equation to make a true sentence. 1. The binomials 3 7 and 3 7 are conjugates. false, 3 7 2. In the expression 45, the radicand is 5. true 3. The sine of an angle is the measure of the opposite leg divided by the measure of the hypotenuse. true 4. The longest side of a right triangle is the hypotenuse. true Lesson-by-Lesson Review 5. After the first step in solving 3x 19 x 3, you would have 3x 19 x2 9. false, 3x 19 x 2 6x 9 6. The two sides that form the right angle in a right triangle are called the legs of the triangle. true 2x3x x 2xy 7. The expression is in simplest radical form. false, y 6y For each lesson, • the main ideas are summarized, • additional examples review concepts, and • practice exercises are provided. 8. A triangle with sides having measures of 25, 20, and 15 is a right triangle. true Vocabulary PuzzleMaker 11-1 Simplifying Radical Expressions See pages 586–592. The Vocabulary PuzzleMaker software improves students’ mathematics vocabulary using four puzzle formats— crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. ELL Example Concept Summary • A radical expression is in simplest form when no radicands have perfect square factors other than 1, no radicands contain fractions, and no radicals appear in the denominator of a fraction. 3 Simplify . 5 2 5 2 to rationalize the denominator. 3 5 2 Multiply by 3 5 2 5 2 5 2 5 2 3(5) 32 2 2 (a b)(a b) a2 b2 15 32 2 2 2 15 32 Simplify. 5 2 MindJogger Videoquizzes 25 2 ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. 23 632 Chapter 11 Radical Expressions and Triangles Round 1 Concepts (5 questions) Round 2 Skills (4 questions) Round 3 Problem Solving (4 questions) www.algebra1.com/vocabulary_review TM Have students flip back through their Foldables to make sure they have included information for every lesson. For more information about Foldables, see Teaching Mathematics with Foldables. 632 rationalizing the denominator (p. 588) similar triangles (p. 616) sine (p. 623) solve a triangle (p. 625) tangent (p. 623) trigonometric ratios (p. 623) Chapter 11 Radical Expressions and Triangles Encourage students to refer to their Foldables while completing the Study Guide and Review and use them in preparing for the Chapter Test. Study Guide and Review Chapter 11 Study Guide and Review Exercises Simplify. See Examples 1–5 on pages 586–589. 60 y 215 y 9 27 92 12. 3 2 7 9. 2 10. 44a2b5 2ab2 11b 11. 3 212 2 57 243 27 13. 3a3b4 a 6 14. 8ab10 4b3 35 53 521 35 3 15 11-2 Operations with Radical Expressions See pages 593–597. Concept Summary • Radical expressions with like radicands can be added or subtracted. • Use the FOIL Method to multiply radical expressions. Examples 1 Simplify 6 54 312 53. 312 53 6 54 6 32 6 3 22 3 53 Simplify radicands. 6 32 6 3 22 3 53 Product Property of Square Roots 6 36 323 53 Evaluate square roots. 6 36 63 53 Simplify. 26 113 Add like radicands. 2 Find 23 5 10 46 . 23 5 10 46 F irst terms Outer terms I nner terms Last terms 23 10 2346 5 10 546 230 818 50 430 Multiply. 230 32 2 52 2 430 8 Prime factorization 230 242 52 430 Simplify. 230 192 Combine like terms. Exercises Simplify each expression. See Examples 1 and 2 on pages 593 and 594. 15. 23 85 35 33 16. 26 48 26 43 17. 427 648 363 18. 47k 77k 27k 7k 19. 518 3112 398 20. 8 15. 53 55 19. 62 127 18 9 2 4 Find each product. See Example 3 on page 594. 24. 1810 30 62 25 21. 2 3 33 32 36 23. 3 2 22 3 6 1 22. 525 7 10 35 24. 65 232 5 Chapter 11 Study Guide and Review 633 Chapter 11 Study Guide and Review 633 Study Guide and Review Chapter 11 Study Guide and Review 11-3 Radical Expressions Answers See pages 598–603. 31. 34 32. 234 11.66 33. 55 11.18 34. 4 195 55.86 35. 24 36. 2 Example Concept Summary • Solve radical equations by isolating the radical on one side of the equation. Square each side of the equation to eliminate the radical. 4x 6 7. Solve 5 5 4x 6 7 5 4x 13 Original equation Add 6 to each side. 5 4x 169 Square each side. 4x 164 Subtract 5 from each side. x 41 Divide each side by 4. Exercises Solve each equation. Check your solution. See Examples 2 and 3 on page 599. no 26 25. 10 2b 0 solution 26. a 4 6 32 27. 7x 1 5 7 28. 4a 20 3 3 29. x 4 x 8 12 30. 3x 14 x6 5 11-4 The Pythagorean Theorem See pages 605–610. Concept Summary • If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then c2 a2 b2. • If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 a2 b2, then the triangle is a right triangle. c a b Example 15 Find the length of the missing side. c2 a2 b2 Pythagorean Theorem 252 152 b2 c 25 and a 15 625 225 Evaluate squares. b2 400 b2 20 b 25 b Subtract 225 from each side. Take the square root of each side. Exercises If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round answers to the nearest hundredth. See Example 2 on page 606. 31. a 30, b 16, c ? 34. b 4, c 56, a ? 31–36. See margin. 32. a 6, b 10, c ? 35. a 18, c 30, b ? 33. a 10, c 15, b ? 36. a 1.2, b 1.6, c ? Determine whether the following side measures form right triangles. See Example 4 on page 608. 37. 9, 16, 20 no 634 634 Chapter 11 Radical Expressions and Triangles Chapter 11 Radical Expressions and Triangles 38. 20, 21, 29 yes 39. 9, 40, 41 yes 40. 18, 24 , 30 no Study Guide and Review Chapter 11 Study Guide and Review 11-5 The Distance Formula See pages 611–615. Concept Summary • The distance d between any two points with coordinates (x1, y1) and (x2, y2) B (x2, y2) (x2 x1)2 (y2 y1)2. is given by d Example y A(x1, y1) O x Find the distance between the points with coordinates (5, 1) and (1, 5). x1)2 (y2 y1)2 d (x 2 (1 (5))2 (5 1)2 Distance Formula (x1, y1) (5, 1) and (x2, y2) (1, 5) Simplify. 36 16 Evaluate squares. 52 or about 7.21 units Simplify. 62 42 42. 58 7.62 43. 205 14.32 Exercises Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. See Example 1 on page 611. 41. (9, 2), (1, 13) 17 42. (4, 2), (7, 9) 43. (4, 6), (2, 7) 44. 25 , 9, 45, 3 45. (4, 8), (7, 12) 46. (2, 6), (5, 11) 214 7.48 137 11.70 74 8.60 Find the value of a if the points with the given coordinates are the indicated distance apart. See Example 3 on page 612. 47. (3, 2), (1, a); d 5 5 or 1 48. (1, 1), (4, a); d 5 5 or 3 49. (6, 2), (5, a); d 145 10 or 14 50. (5, 2), (a, 3); d 170 18 or 8 11-6 Similar Triangles See pages 616–621. Concept Summary • Similar triangles have congruent corresponding angles and proportional corresponding sides. B E • If ABC DEF, then AB BC AC . DE EF DF A C D Example Find the measure of side a if the two triangles are similar. 10 6 5 a Corresponding sides of similar triangles are proportional. 10a 30 Find the cross products. a 3 Divide each side by 10. F 5 cm a cm 10 cm 6 cm Chapter 11 Study Guide and Review 635 Chapter 11 Study Guide and Review 635 • Extra Practice, see pages 844–846. • Mixed Problem Solving, see page 863. Study Guide and Review Exercises For each set of measures given, find the measures of the remaining sides if ABC DEF. See Example 2 on page 617. 51–54. See margin. Answers 45 8 51. 52. 53. 54. 27 4 51. d , e 52. d 9.6, e 7.2 44 53. b , d 6 3 c 16, b 12, a 10, f 9 a 8, c 10, b 6, f 12 c 12, f 9, a 8, e 11 b 20, d 7, f 6, c 15 E d f B D a c A F C b 54. a 17.5, e 8 e 11-7 Trigonometric Ratios See pages 623–630. Concept Summary Three common trigonometric ratios are sine, cosine, and tangent. BC • sin A AB B hypotenuse AC • cos A AB BC • tan A AC Example leg opposite A A A leg adjacent to C Find the sine, cosine, and tangent of A. Round to the nearest ten thousandth. opposite leg hypotenuse 20 or 0.8000 25 sin A adjacent leg hypotenuse 15 or 0.6000 25 A 25 15 cos A C B 20 opposite leg adjacent leg 20 or 1.3333 15 tan A Exercises For ABC, find each value of each trigonometric ratio to the nearest ten thousandth. See Example 1 on page 624. 55. 56. 57. 58. 59. 60. cos B 0.5283 tan A 0.6222 sin B 0.8491 cos A 0.8491 tan B 1.6071 sin A 0.5283 B 28 ft C 53 ft 45 ft A Use a calculator to find the measure of each angle to the nearest degree. See Example 3 on page 624. 61. tan M 0.8043 39° 64. cos F 0.7443 42° 636 636 Chapter 11 Radical Expressions and Triangles Chapter 11 Radical Expressions and Triangles 62. sin T 0.1212 7° 65. sin A 0.4540 27° 63. cos B 0.9781 12° 66. tan Q 5.9080 80° Practice Test Vocabulary and Concepts Assessment Options Match each term and its definition. 1. measure of the opposite side divided by the measure of the hypotenuse b 2. measure of the adjacent side divided by the measure of the hypotenuse a 3. measure of the opposite side divided by the measure of the adjacent side c Vocabulary Test A vocabulary test/review for Chapter 11 can be found on p. 698 of the Chapter 11 Resource Masters. a. cosine b. sine c. tangent Skills and Applications 37 Simplify. 4. 23 9. 3 2 33 6 4. 227 63 43 7. 4 2 10 30 3 3 23 46 3 8. 64 12 46 62 5. 6 6. 112x4y6 4x2y37 9. 1 3 3 2 Solve each equation. Check your solution. 10. 10x 20 40 13. x 6x 8 no solution 11. 4s 1 11 25 14. x 5x 14 7 Form 12. 4x 1 5 6 15. 4x 36x 3 If c is the measure of the hypotenuse of a right triangle, find each missing measure. 16. If necessary, round to the nearest hundredth. 17. 72 8.49 18. 120 10.95 16. a 8, b 10, c ? 17. a 62, c 12, b ? 164 12.81 18. b 13, c 17, a ? Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 20. (1, 1), (1, 5) 210 6.32 21. (9, 2), (21, 7) 19. (4, 7), (4, 2) 9 For each set of measures given, find the measures of the missing sides if ABC JKH. A 64 22. c 20, h 15, k 16, j 12 a 16, b c 65 3 b 23. c 12, b 13, a 6, h 10 j 5, k K 6 24. k 5, c 6.5, b 7.5, a 4.5 h 4.3 , j 3 C B a 3 1 1 1 25. h 1, c 4, k 2, a 3 b 6, j 1 2 2 4 4 Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 26. B 29 A a 21 mA 44° mB 46° a 20 27. C 21 A 15 c B mA 36° 28. B mB 54° c 25.8 a 42˚ 10 C 925 30.41 k j C b H mA 48° a 7.4 b 6.7 29. SPORTS A hiker leaves her camp in the morning. How far is she from camp after walking 9 miles due west and then 12 miles due north? 15 mi C 323 18 units2 D 232 18 units2 232 36 6 www.algebra1.com/chapter_test 1 2A 2B 2C 2D 3 Chapter 11 Tests Type Level Pages MC MC MC FR FR FR basic average average average average advanced 685–686 687–688 689–690 691–692 693–694 695–696 MC = multiple-choice questions FR = free-response questions J h A 30. STANDARDIZED TEST PRACTICE Find the area of the rectangle. B A 16 B 16 2 46 3 18 units2 units2 Chapter Tests There are six Chapter 11 Tests and an OpenEnded Assessment task available in the Chapter 11 Resource Masters. Chapter 11 Open-Ended Assessment Performance tasks for Chapter 11 can be found on p. 697 of the Chapter 11 Resource Masters. A sample scoring rubric for these tasks appears on p. A27. TestCheck and Worksheet Builder This networkable software has three modules for assessment. • Worksheet Builder to make worksheets and tests. • Student Module to take tests on-screen. • Management System to keep student records. Practice Test 637 Portfolio Suggestion Introduction In this chapter, students are introduced to ways in which professionals such as architects and surveyors can use math to solve problems. Ask Students Pick a profession such as architecture, surveying, or engineering and research to find out how triangles, ratios, or trigonometric ratios are used in the field. Write a short report in which you explain how these concepts are used, along with examples. Chapter 11 Practice Test 637 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequently given standardized tests. Part 1 Multiple Choice Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 1. Which equation describes the data in the table? (Lesson 4-8) D A practice answer sheet for these two pages can be found on p. A1 of the Chapter 11 Resource Masters. NAME DATE x y PERIOD Standardized Practice 11 Standardized Test Test Practice Student Recording Sheet, A1Edition.) Student Record Sheet (Use with pages 638–639 of p. the Student 5 2 11 5 1 A 11 B 12 C 22 D 33 4 1 7 A yx6 B y 2x 1 C y 2x 1 D y 2x 1 Part 1 Multiple Choice 6. The function g t2 t represents the total number of games played by t teams in a sports league in which each team plays each of the other teams twice. The Metro League plays a total of 132 games. How many teams are in the league? (Lesson 9-4) B 7. One leg of a right triangle is 4 inches longer than the other leg. The hypotenuse is 20 inches long. What is the length of the shorter leg? (Lesson 11-4) B Select the best answer from the choices given and fill in the corresponding oval. 1 A B C D 4 A B C D 7 A B C D 2 A B C D 5 A B C D 8 A B C D 3 A B C D 6 A B C D 9 A B C D 2. The length of a rectangle is 6 feet more than the width. The perimeter is 92 feet. Which system of equations will determine the length in feet ᐉ and the width in feet w of the rectangle? (Lesson 7-2) C Part 2 Short Response/Grid In Solve the problem and write your answer in the blank. For Questions 13 and 15, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 13 11 12 13 (grid in) 14 15 15 . / . / . . . / . / . . 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 (grid in) 16 Part 3 Quantitative Comparison Select the best answer from the choices given and fill in the corresponding oval. A B C D 19 A B C D 20 A B C wᐉ6 2ᐉ 2w 92 B ᐉw6 ᐉw 92 C ᐉw6 2ᐉ 2w 92 D ᐉw6 ᐉ w 92 D Part 4 Open-Ended Record your answers for Question 21 on the back of this paper. A $450,000 B $734,000 C $900,000 D $1,600,000 B 12 in. 16 in. D 18 in. 8. What is the distance from one corner of the garden to the opposite corner? 5 yd A A 13 yards B 14 yards C 15 yards D 17 yards 12 yd 9. How many points in the coordinate plane are equidistant from both the x- and y-axes and are 5 units from the origin? (Lesson 11-5) D A 0 B 1 C 2 D 4 4. If 32,800,000 is expressed in the form 3.28 10n, what is the value of n? (Lesson 8-3) C Additional Practice See pp. 703–704 in the Chapter 11 Resource Masters for additional standardized test practice. A 5 B 6 C 7 D 8 5. What are the solutions of the equation x2 7x 18 0? (Lesson 9-4) A 638 A 2 or 9 B 2 or 9 C 2 or 9 D 2 or 9 The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit www.princetonreview.com or www.review.com Chapter 11 Radical Expressions and Triangles Test-Taking Tip Questions 7, 21, and 22 Be sure that you know and understand the Pythagorean Theorem. References to right angles, the diagonal of a rectangle, or the hypotenuse of a triangle indicate that you may need to use the Pythagorean Theorem to find the answer to an item. Chapter 11 Radical Expressions and Triangles Log On for Test Practice 638 10 in. (Lesson 11-4) 3. A highway resurfacing project and a bridge repair project will cost $2,500,000 altogether. The bridge repair project will cost $200,000 less than twice the cost of the highway resurfacing. How much will the highway resurfacing project cost? (Lesson 7-2) C 17 18 A Answers 10 A C TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra 1 End-of-Course tests can be found on this CD-ROM. Aligned and verified by Part 2 Short Response/Grid In Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 10. A line is parallel to the line represented by 1 2 3 2 the equation y x 4 0. What is the slope of the parallel line? (Lesson 5-6) 3 Compare the quantity in Column A and the quantity in Column B. Then determine whether: A the quantity in Column A is greater, B the quantity in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. 11. Graph the solution of the system of linear inequalities 2x y 2 and 3x 2y 4. (Lesson 6-6) See margin. 18. 12. The sum of two integers is 66. The second integer is 18 more than half of the first. What are the integers? (Lesson 7-2) 32 and 34 13. The function h(t) 16t2 v0t h0 describes the height in feet above the ground h(t) of an object thrown vertically from a height of h0 feet, with an initial velocity of v0 feet per second, if there is no air friction and t is the time in seconds that it takes the ball to reach the ground. A ball is thrown upward from a 100-foot tower at a velocity of 60 feet per second. How many seconds will it take for the ball to reach the ground? (Lesson 9-5) 5 Column A Column B the value of x in 13x 12 10x 3 the value of y in 12y 16 8y B 19. the slope of 2x 3y 10 the measure of the hypotenuse of a right triangle if the measures of the legs are 10 and 11 the measure of the leg of a right triangle if the measure of the other leg is 13 and the hypotenuse is 390 C x . 16. Simplify the expression x 2 3 x (Lesson 11-1) 3 x 2 or x x 17. The area of a rectangle is 64. The length is x3 x1 , and the width is . What is x? x1 x (Lesson 11-3) 8 or 8 A correct solution that is supported by well-developed, accurate explanations A generally correct solution, but may contain minor flaws in reasoning or computation A partially correct interpretation and/or solution to the problem A correct solution with no supporting evidence or explanation An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given 3 Part 4 Open Ended 1 0 b. To the nearest tenth of a mile, how far (in a straight line) is Haley from her starting point? 11.4 mi c. How did your diagram help you to find Haley’s distance from her starting point? See margin. www.algebra1.com/standardized_test Answers 4 a. Draw a diagram showing the direction and distance of each segment of Haley’s hike. Label Haley’s starting point, her ending point, and the distance, in miles, of each segment of her hike. See margin. 3 3 4 Criteria 2 21. Haley hikes 3 miles north, 7 miles east, and then 6 miles north again. (Lesson 11-4) 3 Score (Lesson 11-4) 7.16 and 0.84 (Lesson 11-1) Sample Scoring Rubric: The following rubric is a sample scoring device. You may wish to add more detail to this sample to meet your individual scoring needs. (Lesson 5-3) Record your answers on a sheet of paper. Show your work. 15. Simplify the expression 381 . Goal: Students use right triangles to find a distance. the y-intercept of 7x 4y 4 B 20. Open-Ended Assessment questions are graded by using a multilevel rubric that guides you in assessing a student’s knowledge of a particular concept. (Lesson 3-5) 14. Find all values of x that satisfy the equation x2 8x 6 0. Approximate irrational numbers to the nearest hundredth. (Lesson 10-4) Evaluating Open-Ended Assessment Questions Part 3 Quantative Comparison 11. Chapter 11 Standardized Test Practice 639 21a. y finish 6 O x 21c. The sketch shows that there are two right triangles. The distance is equal to the sum of the measures of the hypotenuse of each triangle. 7 3 start 7 Chapter 11 Standardized Test Practice 639 Page 604, Follow-Up to Lesson 11-3 Graphing Calculator Investigation 1. {x | x 0}; shifted up 1 6. {x | x 0}; expanded [10, 10] scl: 1 by [10, 10] scl: 1 [10, 10] scl: 1 by [10, 10] scl: 1 7. {x | x 0}; reflected across x-axis Additional Answers for Chapter 11 2. {x | x 0}; shifted down 3 [10, 10] scl: 1 by [10, 10] scl: 1 [10, 10] scl: 1 by [10, 10] scl: 1 8. {x | x 1}; reflected across y-axis, shifted right 1, up 6 3. {x | x 2}; shifted left 2 [10, 10] scl: 1 by [5, 15] scl: 1 [10, 10] scl: 1 by [10, 10] scl: 1 9. {x | x 2.5}; shifted left 2.5, down 4 4. {x | x 5}; shifted right 5 [10, 10] scl: 1 by [10, 10] scl: 1 [5, 15] scl: 1 by [10, 10] scl: 1 5. {x | x 0}; reflected across y-axis [10, 10] scl: 1 by [10, 10] scl: 1 639A Chapter 11 Additional Answers Pages 612–615, Lesson 11-5 44. You can determine the distance between two points by forming a right triangle. Drawing a line through each point parallel to the axes forms the legs of the triangle. The hypotenuse of this triangle is the distance between the two points. You can find the lengths of each leg by subtracting the corresponding x- and y-coordinates, then use the Pythagorean Theorem. Answers should include the following. • You can draw lines parallel to the axes through the two points that will intersect at another point forming a right triangle. The length of a leg of a triangle is the difference in the x- or y-coordinates. The length of the hypotenuse is the distance between the points. Using the Pythagorean Theorem to solve for the hypotenuse, you have the Distance Formula. • The points are on a vertical line so you can calculate distance by determining the absolute difference between the y-coordinates. 57. 3 4 5 6 7 8 9 10 58. 8 7 6 5 4 3 2 1 0 59. 8 7 6 5 4 3 2 1 0 60. 4 3 2 1 0 1 2 3 4 7 8 9 10 61. 2 3 4 5 6 62. 4 3 2 1 0 1 2 3 4 B C A E F D corresponding angles A and D corresponding sides AB and DE B and E C and F BC and E F AC and D F Page 631, Reading Mathematics 1a. Sample answer: A circumstance that brings about a result; any of two or more quantities that form a product when multiplied together; the quantities bring about a result, a product. 1b. Sample answer: The legs of an animal support the animal; one of the two shorter sides of a right triangle; the legs of a triangle support the hypotenuse. 1c. Sample answer: To devise rational explanations for one’s acts without being aware that these are not the real motives; to remove the radical signs from an expression without changing its value; to justify an action without changing its intent. 2a. Sample answer: Rank as determined by the sum of a term’s exponents; the degree of x 2y 2 is 4. 1 of a circle; a semicircle measures 180°. 360 2b. Sample answer: The difference between the greatest and least values in a set of data; the range of 2, 3, 6 is 6 2 or 4. The set of all y-values in a function; the range of {(2, 6), (1, 3)} is {6, 3}. 2c. Sample answer: Circular or spherical; circles are round. To abbreviate a number by replacing its ending digits with zeros; 235,611 rounded to the nearest hundred is 235,600. 3a. Sample answer: dome, domestic, domicile 3b. Sample answer: eradicate, radicand, radius 3c. Sample answer: simile, similarity, similitude Chapter 11 Additional Answers 639B Additional Answers for Chapter 11 2 Pages 618–621, Lesson 11-6 2. Sample answer: ABC DEF