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UNIT 5 Relationships in Geometry MODULE MODULE 15 15 Angles, Triangles, and Equations 6.8.A 16 16 Area and Volume MODULE MODULE Equations 6.8.B, 6.8.C, 6.8.D CAREERS IN MATH Theater Set Construction A person who works in theater set construction works with the set designer to create scenery and needs technical precision when scaling and building sets based on the dimensions of the models. If you are interested in a career in theater set construction, you should study these mathematical subjects: • Geometry • Algebra • Trigonometry At the end of the unit, check out how theater set construction workers use math. © Houghton Mifflin Harcourt Publishing Company • Image Credits: Nik Wheeler/Corbis Research other careers that require technical precision in scaling and building models. Unit 5 Performance Task Unit 5 417 Preview UNIT 5 Vocabulary Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters within found words to answer the riddle at the bottom of the page. B P D Z K M Q Q O Q F W H X Z Y Z J J P H F T S B H Q L Y J W O M W O D H W O L F F G K H P A R A L L E L O G R A M L L R V Q B G L Q Y E S T T A V K E P W N Z R L V K K S R I A Q V N O B I Y I D B U E L X U C L K E G E J T Y Z T L V G P A P L H L G V G K A N E A D L P G T Z W A V S L U E C P Z V I E B B W G C I P B D S M F V Q J L F M B U S M U L O H W J A Z F Z C Q J G C D G S E W P D V C O E S G Q N P C I T D C J S U B M O H R Q B N A M W A L • A triangle that has three congruent sides and three congruent angles. (Lesson 15-2) • A triangle that has two congruent sides. (Lesson 15-3) • A triangle that contains a right angle. (Lesson 15-2) • A quadrilateral where opposite sides are congruent and parallel. (Lesson 16-1) • A quadrilateral in which all sides are congruent and opposite sides are parallel. (Lesson 16-1) Q: Where does a mathematician go when she commits a crime? A: 418 Vocabulary Preview ! © Houghton Mifflin Harcourt Publishing Company • A triangle that has no congruent sides. (Lesson 15-3) Angles, Triangles, and Equations ? MODULE 15 LESSON 15.1 ESSENTIAL QUESTION Determining When Three Lengths Form a Triangle How can you use angles, triangles, and equations to solve real-world problems? 6.8.A LESSON 15.2 Sum of Angle Measures in a Triangle 6.8.A LESSON 15.3 Relationships Between Sides and Angles in a Triangle © Houghton Mifflin Harcourt Publishing Company • Image Credits: Richard Nowitz/ Photodisc/Getty Images 6.8.A Real-World Video You can find examples of triangles all around you. Some buildings, such as the Transamerica Tower, have triangular faces. my.hrw.com my.hrw.com my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 419 Are YOU Ready? Personal Math Trainer Complete these exercises to review skills you will need for this chapter. Inverse Operations EXAMPLE my.hrw.com 7k = 35 Online Assessment and Intervention k is multiplied by 7. To solve the equation, use the inverse operation, division. 35 7k __ = __ 7 7 k=5 7 is added to k. k+7=9 k+7-7=9-7 k= 2 To solve the equation, use the inverse operation, subtraction. Solve each equation using the inverse operation. 1. 9p = 54 2. m - 15 = 9 3. __b8 = 4 4. z + 17 = 23 Name Angles EXAMPLE Use three points of an angle, including the vertex, to name the angle. If there is only one angle at the vertex, you can name the angle by the vertex. Write the vertex between the other two points. ∠AMG, ∠GMA, or ∠M. A M G 5. 6. Z 7. N B S K 420 Unit 5 T J V M R P B T F L © Houghton Mifflin Harcourt Publishing Company Give two names for the angle formed by the dashed rays. Reading Start-Up Vocabulary Review Words ✔ acute angle (ángulo agudo) angle (ángulo) equilateral triangle (triángulo equilátero) inequalities (desigualdad) line segments (segmentos de línea) ✔ obtuse angle (ángulo obtuso) ✔ right angle (ángulo recto) right triangle (triángulo rectángulo) vertex (vértice) Visualize Vocabulary Use the ✔ words to complete the graphic. You will put one word in each oval. Types of Angles Description Angle angle measure > 0° and < 90° angle measure > 90° and < 180° angle measure = 90° Understand Vocabulary Complete the sentences using the review words. 1. A triangle that contains a right angle is a 2. An congruent angles. has three congruent sides and three 3. The sides of triangles are © Houghton Mifflin Harcourt Publishing Company . meet to form an angle of a triangle is called a . Where two lines . Active Reading Pyramid Before beginning the module, create a pyramid to help you organize what you learn. Label each side with one of the lesson titles from this module. As you study each lesson, write important ideas like vocabulary, properties, and formulas on the appropriate side. Module 15 421 MODULE 15 Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module. 6.8.A Extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle. What It Means to You You will learn to determine if three lengths can form a triangle. UNPACKING EXAMPLE 6.8.A A map of a new dog park shows that it is triangular and that the sides measure 18 yd, 37 yd, and 17 yd. Are the dimensions possible? Explain your reasoning. Find the sum of the lengths of each pair of sides. Compare the sum to the third side. ? ? ? 18 + 37 > 17 18 + 17 > 37 37 + 17 > 18 55 > 17 ✔ 35 ≯ 34 ✘ 54 > 18 ✔ 6.8.A Extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle. What It Means to You You will learn how to find the measure of an angle of a triangle if you know the measures of the other two angles. The measures of two of the angles of a triangle are 47° and 81°. What is the measure of the third angle of the triangle? x 47° + 81° + 128° Visit my.hrw.com to see all the unpacked. my.hrw.com 422 Unit 5 C m∠A + m∠B + m∠C = 180° + x = 180° x = 180° x = 52° 47° A The third angle of the triangle measures 52°. 81° B © Houghton Mifflin Harcourt Publishing Company • Image Credits: Thinkstock/ Comstock Images/Getty Images The sum of two of the given lengths is not greater than the third length. So, the dog park cannot have these side lengths. LESSON 15.1 ? Determining When Three Lengths Form a Triangle ESSENTIAL QUESTION Expressions, equations, and relationships— 6.8.A Extend previous knowledge of triangles and their properties to include . . . determining when three lengths form a triangle. How can you use the relationship between side lengths to determine when three lengths form a triangle? EXPLORE ACTIVITY 6.8.A Drawing Three Sides Use geometry software to draw a triangle whose sides have the following lengths: 2 units, 3 units, and 4 units. E F c=4 A Draw three line segments of 2, 3, and 4 units of length. C D b=3 A B a=2 ___ B Let AB be the base of the triangle. Place endpoint C on top of endpoint B and endpoint E on top of endpoint A. These will become two of the vertices of the triangle. F D c=4 E B © Houghton Mifflin Harcourt Publishing Company A a=2 C C Using the endpoints C and E as fixed vertices, rotate endpoints F and D to see if they will meet in a single point. The line segments of 2, 3, and 4 units do / do not form a triangle. D Repeat Steps 2 and 3, but start with a different base length. Do the line segments make the exact same triangle as the original? D c=4 E A a=2 C b=3 F b=3 B The line segments do / do not make the same triangle as the original. E Draw three line segments of 2, 3, and 6 units. Can you form a triangle with the given segments? The line segments of 2, 3, and 6 units do / do not form a triangle. Lesson 15.1 423 EXPLORE ACTIVITY (cont’d) Reflect 1. Conjecture Try to make triangles using real world objects such as three straws of different lengths. Find three side lengths that form a triangle and three side lengths that do not form a triangle. What do you notice about the lengths that do not form a triangle? Using Triangle Side Length Relationships Math On the Spot You saw in the Explore Activity that you cannot always form a triangle from three given line segments. my.hrw.com Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 4 5 7 Can form a triangle 4 2 7 Cannot form a triangle EXAMPLE 1 6.8.A Tell whether a triangle can have sides with the given lengths. Animated Math my.hrw.com A 11 cm, 6 cm, 13 cm STEP 1 Find the sum of the lengths of each pair of sides. ? ? ? 11 + 6 > 13 6 + 13 > 11 11 + 13 > 6 STEP 2 Compare the sum to the third side. 17 > 13 ✓ 19 > 11 ✓ 24 > 6 ✓ The sum of any two of the given lengths is greater than the third length. So, a triangle can have these side lengths. 424 Unit 5 © Houghton Mifflin Harcourt Publishing Company You can use this relationship to determine if given side lengths can form a triangle. B 5 ft, 15 ft, 9 ft STEP 1 Find the sum of the lengths of each pair of sides. ? ? ? 15 + 9 > 5 5 + 9 > 15 5 + 15 > 9 STEP 2 Compare the sum to the third side. 20 > 9 ✓ 24 > 5 ✓ Math Talk 14 ≯ 15 Mathematical Processes Explain why a triangle with sides measuring 5 in., 5 in., and 1 foot cannot be constructed. The sum of any two of the given lengths is not greater than the third length. So, a triangle cannot have these side lengths. YOUR TURN Tell whether a triangle can have sides with the given lengths. Explain. 2. 3 cm, 6 cm, 9 cm 3. Personal Math Trainer 4 m, 5 m, 8 m Online Assessment and Intervention my.hrw.com © Houghton Mifflin Harcourt Publishing Company Using Inequalities to Represent the Relationship Between Triangle Side Lengths Math On the Spot You can use what you know about the relationship among the lengths of the sides of a triangle to write an inequality. Then you can use the inequality to determine if a given value can be the length of an unknown side. EXAMPL 2 EXAMPLE 6.8.A Which value could be the length of x? x = 15 x = 10 4 9 STEP 1 4+9>x 4+9>x x Write an inequality. STEP 2 ? 4 + 9 > 15 ? 4 + 9 > 10 Substitute each value for x. STEP 3 my.hrw.com Math Talk Mathematical Processes Explain how you know that the Triangle Inequality relationship is true for every equilateral triangle. Compare the sum to the given value of x. The value that could be the length of x is x = 10. 13 ≯ 15 13 > 10 ✓ Lesson 15.1 425 YOUR TURN Personal Math Trainer 4. Which value could be the length of x? Online Assessment and Intervention 20 x = 35 13 x = 13 x my.hrw.com Guided Practice Determine whether a triangle can have sides with the given lengths. Explain. (Explore Activity and Example 1) 1. 3 cm, 10 cm, 8 cm 2. 10 ft, 10 ft, 18 ft 3. 30 in., 20 in., 40 in. 4. 16 cm, 12 cm, 3 cm x = 29 x = 45 17 x 22 ? ? ESSENTIAL QUESTION CHECK-IN 6. Explain how you can determine whether three metal rods can be joined to form a triangle. 426 Unit 5 © Houghton Mifflin Harcourt Publishing Company 5. Which value could be the length of x? (Example 2) Name Class Date 15.1 Independent Practice Personal Math Trainer 6.8.A 7. A map of a new dog park shows that it is triangular and that the sides measure 18.5 m, 36.9 m, and 16.9 m. Are the dimensions correct? Explain your reasoning. my.hrw.com Online Assessment and Intervention 10. Geography The map shows the distance in air miles from Houston to both Austin and San Antonio. Austin 146.43 mi Houston San Antonio 8. Choose a real world object that you can cut into three different lengths to form a triangle. Find three side lengths that form a triangle and three lengths that do not form a triangle. For each triangle, give the side lengths and explain why those lengths do or do not form a triangle. 189.34 mi a. What is the greatest possible distance from Austin to San Antonio? b. How did you find the answer? Triangle 1: © Houghton Mifflin Harcourt Publishing Company c. What is the least possible distance from Austin to San Antonio? Triangle 2: d. How did you find the answer? 9. Could the three sides of a triangular shopping mall measure _12 mi, _13 mi, and _1 mi? Show how you found your answer. 4 Lesson 15.1 427 11. Critical Thinking Two sides of an isosceles triangle measure 3 inches and 13 inches respectively. Find the length of the third side. Explain your reasoning. FOCUS ON HIGHER ORDER THINKING Work Area 12. Critique Reasoning While on a car trip with her family, Erin saw a sign that read, “Amarillo 100 miles, Lubbock 80 miles.” She concluded that the distance from Amarillo to Lubbock is 100 - 80 = 20 miles. Was she right? Explain. 14. Persevere in Problem Solving A metalworker cut an 8-foot length of pipe into three pieces and welded them to form a triangle. Each of the 3 sections measured a whole number of feet in length. How long was each section? Explain your reasoning. 428 Unit 5 © Houghton Mifflin Harcourt Publishing Company 13. Make a Conjecture Is there a value of n for which there could be a triangle with sides of length n, 2n, and 3n? Explain. LESSON 15.2 ? Sum of Angle Measures in a Triangle ESSENTIAL QUESTION Expressions, equations, and relationships—6.8.A Extend previous knowledge of triangles and their properties to include the sum of angles in a triangle … How do you use the sum of angles in a triangle to find an unknown angle measure? EXPLORE ACTIVITY 6.8.A Exploring Angles in a Triangle Recall that a triangle is a closed figure with three line segments and three angles. The measures of the angles of a triangle have a special relationship with one another. 2 A Use a straightedge to draw a large triangle. Label the angles 1, 2, and 3. 3 1 B Use scissors to cut out the triangle. C Tear off the three angles. Arrange them around a point on a line as shown. 1 2 3 D What is the measure of the straight angle formed by the three angles? © Houghton Mifflin Harcourt Publishing Company E What is the sum of the measures of the three angles? Explain. F Compare your results with those of your classmates. What guess can you make? Reflect 1. Justify Reasoning How can you show that your guess is correct? Lesson 15.2 429 Finding an Angle Measure in a Triangle Math On the Spot Sum of Angle Measures of a Triangle my.hrw.com The sum of the measures of the angles in a triangle is 180°. 2 1 3 m∠1 + m∠2 + m∠3 = 180° EXAMPLE 1 6.8.A Fountain Place, shown to the right, is a 720-foot Dallas skyscraper. Find the measure of the unknown angle in the triangle at the top of the building. The sum of the angle measures m∠1 + m∠2 + m∠3 = 180° in a triangle is 180°. x 65° + 65° + x = 180° Write an equation. 130° + x = 180° −130° 65° Add. −130° 65° Subtract 130° from both sides. x = 50° The angle at the top of the triangle measures 50°. Math Talk Mathematical Processes © Houghton Mifflin Harcourt Publishing Company Can a triangle have two obtuse angles? Why or why not? YOUR TURN Find the unknown angle measures. 2. D 100° x E K 3. x 55° F Personal Math Trainer Online Assessment and Intervention my.hrw.com 430 Unit 5 J x= x= 71° 56° L Finding Angles in an Equilateral Triangle Recall that an equilateral triangle has three congruent sides and three congruent angles. Math On the Spot my.hrw.com EXAMPL 2 EXAMPLE 6.8.A Find the angle measures in the equilateral triangle. 3x = 180° Write an equation. 3x = ____ 180° ___ Divide both sides by 3. 3 3 x = 60° x x x Each angle in an equilateral triangle measures 60°. Reflect 4. Multiple Representations Write a different equation to find the angle measures in Example 2. Will the answer be the same? Explain. © Houghton Mifflin Harcourt Publishing Company 5. Draw Conclusions Triangle ABC is a right triangle. What conclusions can you draw about the measures of the angles of the triangle? YOUR TURN Write an equation to find the unknown angle measure in each triangle. 6. The measures of two of the angles are 25° and 65°. 7. The measures of two of the angles are 60°. 8. The measures of two of the angles are 35°. Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 15.2 431 Guided Practice 1. The sum of the angle measures in a triangle is (Explore Activity) . Find the unknown angle measure in each triangle. (Examples 1 and 2) 2. m∠R + m∠S + m∠T = + S 105° +x= T x +x= - 42° - R x= K 3. A x 4. 96° x 42° L 28° C x= x= 5. G M 6. 61° 33° F x 28° H x= P x 8. The measures of two of the angles are 50° and 30°. ESSENTIAL QUESTION CHECK-IN 9. Arlen knows the measures of two angles of a triangle. Explain how he can find the measure of the third angle. Why does your method work? 432 Unit 5 59° N x= 7. The measures of two of the angles are 45°. ? ? B © Houghton Mifflin Harcourt Publishing Company J Name Class Date 15.2 Independent Practice Personal Math Trainer 6.8.A Figure ABCD represents a garden crossed by ___ straight walkway AC. Use the figure for 10–15. A 16. An observer at point O sees airplane P directly over airport A. The observer measures the angle of the plane at 40.5°. B P 57° 100° D Online Assessment and Intervention my.hrw.com 32° 88° C O 10. Find m∠DAC. 40.5° A Find m∠P. 11. Explain how you found m∠DAC. The map shows the intersection of three streets in San Antonio’s River Walk district. Use the map for 17–18. B 12. Find m∠BAC. N ll S 48° © Houghton Mifflin Harcourt Publishing Company 13. Explain how you found m∠BAC. A h y’ n C 17. Find the measures of the three angles of the triangle. 14. Find m∠DAB. 15. Explain how you found m∠DAB. 18. Explain how you found the angle measures. Lesson 15.2 433 FOCUS ON HIGHER ORDER THINKING Work Area 19. Persevere in Problem Solving Find the measure of ∠ACB. Explain how you found your answer. A 83° x D 148° B C 20. Communicate Mathematical Ideas Explain how you can use the figure to find the sum of the measures of the angles of quadrilateral ABCD. What is the sum? B A C 21. Draw Conclusions Recall that a right triangle is a triangle with one right angle. One angle of a triangle measures 89.99 degrees. Can the triangle be a right triangle? Explain your reasoning. 434 Unit 5 © Houghton Mifflin Harcourt Publishing Company D LESSON 15.3 ? Relationships Between Sides and Angles in a Triangle ESSENTIAL QUESTION EXPLORE ACTIVITY Expressions, equations, and relationships—6.8.A Extend previous knowledge of triangles and their properties to include…the relationship between the lengths of sides and measures of angles in a triangle… How can you use the relationships between side lengths and angle measures in a triangle to solve problems? 6.8.A Exploring the Relationship Between Sides and Angles in a Triangle There is a special relationship between the lengths of sides and the measures of angles in a triangle. A Use geometry software to make triangle ABC. Make ∠A the smallest angle. B C A B Choose one vertex and drag it so that you lengthen the side of the triangle opposite angle A. Describe what happens to ∠A. B C © Houghton Mifflin Harcourt Publishing Company A C Drag the vertex to shorten the side opposite ∠B. What happens to ∠B? D Make several new triangles. In each case, note the locations of the longest and shortest sides in relation to the largest and smallest angles. Describe your results. B A C Lesson 15.3 435 Using the Relationship Between Sides and Angles in a Triangle my.hrw.com EXAMPLE 1 My Notes 6.8.A A Triangle ABC has side lengths of 7 cm, 9 cm, and 4.5 cm. Use the relationship between the sides and angles of a triangle to match each side with its correct length. B 100° A 50° 30° AC = 9 cm The longest side is opposite the largest angle. AB = 4.5 cm The shortest side is opposite the smallest angle. BC = 7 cm The midsize side is opposite the midsize angle. B Triangle ABC has angles measuring 60°, 80°, and 40°. Use the relationship between the sides and angles of a triangle to match each angle with its correct measure. B 20 13 m∠C = 40° A The largest angle is opposite the 17.6 longest side. The smallest angle is opposite the shortest side. m∠B = 60° The midsize angle is opposite the midsize side. m∠A = 80° YOUR TURN 1. 2. Personal Math Trainer Online Assessment and Intervention my.hrw.com 436 Unit 5 AC = BC = A 88° 57° 35° m∠B = 6 m∠C = A C C Triangle ABC has angle measures of 45°, 58°, and 77°. Match each angle with its correct measure. m∠A = C B Triangle ABC has side lengths of 11, 16, and 19. Match each side with its correct length. AB = C 5 7 B © Houghton Mifflin Harcourt Publishing Company Math On the Spot You have seen that in a triangle the largest angle is opposite the longest side and the smallest angle is opposite the shortest side. It follows that the midsize angle is opposite the midsize side. Solving Problems Using Triangle Relationships Recall that triangles can be classified by the lengths of their sides. A scalene triangle has no congruent sides. An isosceles triangle has two congruent sides. An equilateral triangle has three congruent sides. EXAMPL 2 EXAMPLE Problem Solving FPO Math On the Spot my.hrw.com 6.8.A Brandy is making a quilt. Each block of the quilt is made up of four triangles. Each triangle is in the shape of a right isosceles triangle. Two of the side measures of one triangle are 6.4 inches and 9 inches. Brandy wants to add a ribbon border around one of the triangles. How much ribbon will she need? Analyze Information Rewrite the question as a statement. • Find the amount of ribbon Brandy will need for a border around one triangle. Identify the important information. • Each quilt piece has the shape of a right isosceles triangle. • Two sides of the triangle measure 6.4 inches and 9 inches. Formulate a Plan You can draw a model and label it with the important information to find the total length of ribbon that Brandy needs for one triangle. © Houghton Mifflin Harcourt Publishing Company Justify and Evaluate Solve Think: A right triangle will have one 90° angle. Since the sum of the angles is 180°, the other two angles will be congruent and will have a combined measure of 90°. 90° ÷ 2 = 45° Label the new information on the model. 90° is the greatest angle measure, so the side opposite the 90° angle will be the longest side. The other two angles are congruent, so the sides opposite those angles are congruent. The shortest side lengths are 6.4 inches and 6.4 inches. So, Brandy will need 6.4 + 6.4 + 9 = 21.8 inches of ribbon. 45° 90° 45° 90° 45° longest side 45° Justify and Evaluate The solution is reasonable because the quilt piece is in the shape of an isosceles right triangle and it has two sides measuring 6.4 inches and 9 inches. Lesson 15.3 437 YOUR TURN 3. Personal Math Trainer Online Assessment and Intervention A fence around a rock garden is in the shape of a right triangle. Two angles measure 30° and 60°. Two sides measure 10 feet and 17.3 feet. The total length of the fence is 47.3 feet. How long is the side opposite the right angle? my.hrw.com Guided Practice B 1. Triangle ABC has side lengths of 17, 13, and 24. Match each side with its correct length. (Example 1) = 24 = 13 = 17 2. The figure represents a traffic island that has angles measuring 60°, 20°, and 100°. Match each angle with its correct measure. (Example 1) m∠ = 100° m∠ = 20° m∠ 105° A = 60° 3. Vocabulary Explain how the relationship between the sides and angles of a triangle applies to equilateral triangles. (Example 2) 43° 32° C N 58 in. 20 in. M 51 in. P ? ? ESSENTIAL QUESTION CHECK-IN 5. Describe the relationship between the lengths of the sides and the measures of the angles in a triangle. 438 Unit 5 © Houghton Mifflin Harcourt Publishing Company 4. Ramone is building a fence around a vegetable garden in his backyard. The fence will be in the shape of a right isosceles triangle. Two of the side measures are 12 feet and 16 feet. Use a problem solving model to find the total length of fencing he needs. Explain. (Example 2) Name Class Date 15.3 Independent Practice Personal Math Trainer 6.8.A my.hrw.com Use the figure for 6–8. Online Assessment and Intervention The figure shows the angle measurements formed by two fenced-in animal pens that share a side. Use the figure for 9–10. G B A 57° 58° 58° F 58.5° 61° H 6. Critique Reasoning Dustin says that △FGH is an equilateral triangle because the sides appear to be the same length. Is his reasoning valid? Explain. 68° 54° D 65° C _ 9. Caitlin says that AC is the longest segment of fencing because it is opposite 68°, the largest angle measure in the figure. Is her reasoning valid? Explain. © Houghton Mifflin Harcourt Publishing Company 7. What additional information do you need to know before you can determine which side of the triangle is the longest? How can you find it? 10. What is the longest segment of fencing in △ABC? Explain your reasoning. 8. Which side of the triangle is the longest? Explain how you found the answer. 11. Find the longest segment of fencing in the figure. Explain your reasoning. Lesson 15.3 439 _ _ _ _ 12. In triangle ABC, AB is longer than BC and BC is longer than AC. a. Draw a sketch of triangle ABC. b. Name the smallest angle in the triangle. Explain your reasoning. FOCUS ON HIGHER ORDER THINKING Work Area Z 13. Persevere in Problem Solving Determine the shortest line segment in the figure. Explain how you found the answer. 30° 30° X 14. Communicate Mathematical Ideas Explain how the relationship between the sides and angles of a triangle applies to isosceles triangles. 15. Critical Thinking Can a scalene triangle contain a pair of congruent angles? Explain. 440 Unit 5 W © Houghton Mifflin Harcourt Publishing Company Y MODULE QUIZ Ready Personal Math Trainer 15.1 Determining When Three Lengths Form a Triangle Online Assessment and Intervention Determine whether the three side lengths form a triangle. my.hrw.com 1. 3, 5, 7 2. 9, 15, 4 3. 17, 5, 23 4. 28, 16, 38 15.2 Sum of Angle Measures in a Triangle Find the unknown angle measures. 5. 6. 88° 41° 38° 112° 15.3 Relationships Between Sides and Angles in a Triangle Match each of the given measures to the correct side or angle. 7. 11, 7.5, 13 8. 24°, 44°, 112° A 89° 16 © Houghton Mifflin Harcourt Publishing Company D 35° C 7 12 56° B E F ESSENTIAL QUESTION 9. How can you describe the relationships among angles and sides in a triangle? Module 15 441 Personal Math Trainer MODULE 15 MIXED REVIEW Texas Test Prep Selected Response 1. The two longer sides of a triangle measure 16 and 22. Which of the following is a possible length of the shortest side? A 4 C B 6 D 19 my.hrw.com 5. Which of these could be the value of x in the triangle below? A 50° 11 29 2. Part of a large metal sculpture will be a triangle formed by welding three bars together. The artist has four bars that measure 12 feet, 7 feet, 5 feet, and 3 feet. Which bar could not be used with two of the others to form a triangle? 4x 43° C 87° 22 B A 5 B 6 A the 3-foot bar C B the 5-foot bar C Online Assessment and Intervention 7 D 10 the 7-foot bar D the 12-foot bar 3. What is the measure of the missing angle in the triangle below? Gridded Response 6. Find m∠Z. X F 133° 29° Y . D 65° 56° A 39° C B 49° D 69° E 59° 4. The measure of ∠A in △ABC is 88°. The measure of ∠B is 60% of the measure of ∠A. What is the measure of ∠C? 442 A 39.2° C B 52.8° D 127.2° Unit 5 91° 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 © Houghton Mifflin Harcourt Publishing Company Z Area and Volume Equations ? MODULE 16 LESSON 16.1 ESSENTIAL QUESTION Area of Quadrilaterals How can you use area and volume equations to solve real-world problems? 6.8.B, 6.8.D LESSON 16.2 Area of Triangles 6.8.B, 6.8.D LESSON 16.3 Solving Area Equations 6.8.C, 6.8.D LESSON 16.4 Solving Volume Equations © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Getty Royalty Free 6.8.C, 6.8.D Real-World Video my.hrw.com my.hrw.com Quilting, painting, and other art forms use familiar geometric shapes, such as triangles and rectangles. To buy enough supplies for a project, you need to find or estimate the areas of each shape in the project. my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 443 Are YOU Ready? Personal Math Trainer Complete these exercises to review skills you will need for this chapter. my.hrw.com Use of Parentheses EXAMPLE Online Assessment and Intervention 1 _ (14) (12 + 18) = _12 (14) (30) Perform operations inside parentheses first. 2 = 7 (30) Multiply left to right. = 210 Multiply again. Evaluate. 1. _12 (3) (5 + 7) 2. _12 (15) (13 + 17) 3. _12 (10) (9.4 + 3.6) 4. _12 (2.1) (3.5 + 5.7) Area of Square, Rectangles, Triangle EXAMPLE Find the area of the rectangle. 8 ft 3 ft A = bh Use the formula for area of a rectangle. = 8 · 3 Substitute for base and height. = 24 Multiply. Area equals 24 square feet. Find the area of each figure. 6. a square with sides of 7.6 m 7. a rectangle with length 3_41 ft and width 2_21 ft 8. a triangle with base 8.2 cm and height 5.1 cm 444 Unit 5 © Houghton Mifflin Harcourt Publishing Company 5. a triangle with base 6 in. and height 3 in. Reading Start-Up Visualize Vocabulary Use the ✔ words to complete the graphic. You will put one word in each oval. Types of Triangles has three congruent sides and three congruent angles contains a 90° angle has two equal sides and two equal angles Vocabulary Review Words base (base) ✔ equilateral triangle (triángulo equilátero) ✔ isosceles triangle (triángulo isósceles) legs (catetos) quadrilateral (cuadrilátero) rectangular prism (prisma rectangular) ✔ right triangle (triángulo rectángulo) volume (volumen) Preview Words parallelogram (paralelogramo) rhombus (rombo) trapezoid (trapecio) Understand Vocabulary © Houghton Mifflin Harcourt Publishing Company Match the term on the left to the correct expression on the right. 1. parallelogram A. A quadrilateral in which all sides are congruent and opposite sides are parallel. 2. trapezoid B. A quadrilateral in which opposite sides are parallel and congruent. 3. rhombus C. A quadrilateral in which two sides are parallel. Active Reading Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and formulas. Refer to your finished booklet as you work on assignments and study for tests. Module 16 445 MODULE 16 Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module. 6.8.C Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. What It Means to You You will use the formula for the area of a figure to write an equation that can be used to solve a problem. UNPACKING EXAMPLE 6.8.C The Hudson Middle School wrestling team won the state tournament and was awarded a triangular pennant to display in the school gymnasium. The pennant has an area of 2.25 square meters. The base of the pennant is 1.5 meters long. Write an equation to find the height of the pennant. A = _12 bh 2.25 = _12 (1.5)h 2.25 = 0.75h 1.5 m 6.8.D Determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. What It Means to You You will use the formula for the volume of a rectangular prism. UNPACKING EXAMPLE 6.8.D Jala has an aquarium in the shape of a rectangular prism with a volume of 2,160 cubic inches. The length is 15 inches and the width is 12 inches. Find the height of the aquarium. v=l·w·h 2,160 = 15 · 12 · h 2,160 = 180 · h 2,160 _____ =h 180 Visit my.hrw.com to see all the unpacked. my.hrw.com 446 Unit 5 12 = h The height of the aquarium is 12 inches. © Houghton Mifflin Harcourt Publishing Company • Image Credits: GK Hart/Vikki Hart/Getty Images An equation to find the height of the pennant is 2.25 = 0.75h. LESSON 16.1 ? Area of Quadrilaterals ESSENTIAL QUESTION Expressions, equations, and relationships— 6.8.B Model area formulas for parallelograms, trapezoids ... by decomposing and rearranging parts of these shapes. Also 6.8.D How can you find the areas of parallelograms, rhombuses, and trapezoids? EXPLORE ACTIVITY 6.8.B Area of a Parallelogram Recall that a rectangle is a special type of parallelogram. A Draw a large parallelogram on grid paper. Cut out your parallelogram. B Cut your parallelogram on the dashed line as shown. Then move the triangular piece to the other side of the parallelogram. height (h) base (b) width (w) length (l) C What figure have you formed? Does this figure have the same area as the parallelogram? © Houghton Mifflin Harcourt Publishing Company base of parallelogram = of rectangle height of parallelogram = area of parallelogram = of rectangle Math Talk of rectangle Mathematical Processes What is the formula for the area of this figure? A = or D What is the formula for the area of a parallelogram? A = How is the relationship between the length and width of a rectangle similar to the relationship between the base and height of a parallelogram? Area of a Parallelogram The area A of a parallelogram is the product of its base b and its height h. A = bh h b Lesson 16.1 447 EXPLORE ACTIVITY (cont’d) Reflect 1. Find the area of the parallelogram. 7 cm A= 14 cm Finding the Area of a Trapezoid Math On the Spot To find the formula for the area of a trapezoid, notice that two copies of the same trapezoid fit together to form a parallelogram. Therefore, the area of the trapezoid is _12 the area of the parallelogram. my.hrw.com h b A= The height of the parallelogram is the same as the height of the trapezoid. The base of the parallelogram is the sum of the two bases of the trapezoid. b ↓ ·h A = (b1 + b2) · h Area of a Trapezoid b1 h b2 EXAMPLE 1 Math Talk A section of a deck is in the shape of a trapezoid. What is the area of this section of the deck? Mathematical Processes Does it matter which of the trapezoid’s bases is substituted for b1 and which is substituted for b2? Why or why not? Animated Math my.hrw.com 448 Unit 5 6.8.D 17 ft b1 = 17 b2 = 39 h = 16 Use the formula for area of a trapezoid. 1 h(b + b ) A = __ 2 2 1 1 · 16(17 + 39) = __ 2 1 · 16(56) = __ 2 16 ft 39 ft Substitute. Add inside the parentheses. = 8 · 56 1 and 16. Multiply __ 2 = 448 square feet Multiply. © Houghton Mifflin Harcourt Publishing Company The area of a trapezoid is half its height multiplied by the sum of the lengths of its two bases. 1 h(b + b ) A = __ 2 2 1 YOUR TURN 2. Another section of the deck is also shaped like a trapezoid. For this section, the length of one base is 27 feet, and the length of the other base is 34 feet. The height is 12 feet. What is the area of this section of the deck? A = f t2 Personal Math Trainer Online Assessment and Intervention my.hrw.com Finding the Area of a Rhombus A rhombus is a quadrilateral in which all sides are congruent and opposite sides are parallel. A rhombus can be divided into four triangles that can then be rearranged into a rectangle. Math On the Spot my.hrw.com h b A=b · h ↓ ↓ The base of the rectangle is the same length as one of the diagonals of the rhombus. The height of the rectangle is _21 the length of the other diagonal. A = d1 · _12 d2 Area of a Rhombus The area of a rhombus is half of the product of its two diagonals. © Houghton Mifflin Harcourt Publishing Company A = _12 d1d2 d2 d1 EXAMPL 2 EXAMPLE 6.8.B Cedric is constructing a kite in the shape of a rhombus. The spars of the kite measure 15 inches and 24 inches. How much fabric will Cedric need for the kite? To determine the amount of fabric needed, find the area of the kite. d1 = 15 15 in. d2 = 24 Use the formula for area of a rhombus. A = _12 d1d2 24 in. = _12 (15)(24) Substitute. = 180 square inches Multiply. Lesson 16.1 449 YOUR TURN Find the area of each rhombus. Personal Math Trainer 3. d1 = 35 m; d2 = 12 m Online Assessment and Intervention A= 4. d1 = 9.5 in.; d2 = 14 in. A= m2 in2 my.hrw.com 5. d1 = 10 m; d2 = 18 m A= 6. d1 = 8_14 ft; d2 = 40 ft A= m2 ft2 Guided Practice 1. Find the area of the parallelogram. (Explore Activity) A = bh =( 9 in. )( = ) 13 in. in.2 2. Find the area of the trapezoid. (Example 1) A = _12h(b1 + b2) = _12 ( = )( + 9 cm ) 14 cm cm2 15 cm A = _12 d1d2 = _12 ( = ? ? )( ) 18 in. 11 in. in.2 ESSENTIAL QUESTION CHECK-IN 4. How can you find the areas of parallelograms, rhombuses, and trapezoids? 450 Unit 5 © Houghton Mifflin Harcourt Publishing Company 3. Find the area of the rhombus. (Example 2) Name Class Date 16.1 Independent Practice Personal Math Trainer 6.8.B, 6.8.D 5. Rearrange the parts of the parallelogram to form a rectangle. Find the area of the parallelogram and the area of the rectangle. What is the relationship between the areas? my.hrw.com Online Assessment and Intervention 9. The seat of a bench is in the shape of a trapezoid with bases of 6 feet and 5 feet and a height of 1.5 feet. What is the area of the seat? 6 cm 10. A kite in the shape of a rhombus has diagonals that are 25 inches long and 15 inches long. What is the area of the kite? 14 cm 6. What is the area of a parallelogram that has a base of 12_34 in. and a height of 2_12 in.? 7. Draw a copy of the trapezoid to form a parallelogram. Find the area of the trapezoid and the area of the parallelogram. What is the relationship between the areas? © Houghton Mifflin Harcourt Publishing Company 42 in. 11. A window in the shape of a parallelogram has a base of 36 inches and a height of 45 inches. What is the area of the window? 12. Communicate Mathematical Ideas Find the area of the figure. Explain how you found your answer. 10 ft 6 ft 24 in. 12 ft 36 in. 18 ft 8. The bases of a trapezoid are 11 meters and 14 meters. Its height is 10 meters. What is the area of the trapezoid? Lesson 16.1 451 13. Multistep A parking space shaped like a parallelogram has a base of 17 feet and a height is 9 feet. A car parked in the space is 16 feet long and 6 feet wide. How much of the parking space is not covered by the car? FOCUS ON HIGHER ORDER THINKING Work Area 14. Critique Reasoning Simon says that to find the area of a trapezoid, you can multiply the height by the top base and the height by the bottom base. Then add the two products together and divide the sum by 2. Is Simon correct? Explain your answer. 16. Critique Reasoning Find the area of the trapezoid using the formula A = _12 h(b1 + b2). Decompose the trapezoid into a rectangle and a triangle and find the area of each. Then find the sum of the two areas. Compare this sum with the area of the trapezoid. 452 Unit 5 12 cm 8 cm 6 cm © Houghton Mifflin Harcourt Publishing Company 15. Multistep The height of a trapezoid is 8 in. and its area is 96 in2. One base of the trapezoid is 6 inches longer than the other base. What are the lengths of the bases? Explain how you found your answer. LESSON 16.2 Area of Triangles ? Expressions, equations, and relationships— 6.8.B Model area formulas for ... triangles by decomposing and rearranging parts of these shapes. Also 6.8.D ESSENTIAL QUESTION How do you find the area of a triangle? EXPLORE ACTIVITY 1 6.8.B h Area of a Right Triangle b A Draw a large rectangle on grid paper. What is the formula for the area of a rectangle? A = B Draw one diagonal of your rectangle. The diagonal divides the rectangle into Each one represents . of the rectangle. Use this information and the formula for area of a rectangle to write a formula for the area of a right triangle. A = Reflect © Houghton Mifflin Harcourt Publishing Company 1. Communicate Mathematical Ideas In the formula for the area of a right triangle, what do b and h represent? EXPLORE ACTIVITY 2 6.8.B Area of a Triangle A Draw a large triangle on grid paper. Do not draw a right triangle. B Cut out your triangle. Then trace around it to make a copy of your triangle. Cut out the copy. C Cut one of your triangles into two pieces by cutting through one angle directly across to the opposite side. Now you have three triangles — one large triangle and two smaller triangles. Lesson 16.2 453 EXPLORE ACTIVITY 2 (cont’d) When added together, the areas of the two smaller triangles equal the of the large triangle. D Arrange the three triangles into a rectangle. What fraction of the rectangle does the large h triangle represent? The area of the rectangle is A = bh. What is the area b of the large triangle? A = How does this formula compare to the formula for the area of a right triangle that you found in Explore Activity 1? Reflect 2. Communicate Mathematical Ideas What type of angle is formed by the base and height of a triangle? Finding the Area of a Triangle Math On the Spot The area A of a triangle is half the product of its base b and its height h. my.hrw.com h A = _12 bh b EXAMPLE 1 6.8.D Find the area of each triangle. b = 20 meters A 8m 20 m 454 Unit 5 h = 8 meters A = _12 bh = _12 (20 meters) (8 meters) Substitute. = 80 square meters Multiply. © Houghton Mifflin Harcourt Publishing Company Area of a Triangle Find the area of each triangle. b = 12 inches B h = 5 inches A = _12 bh 5 in. = _12 (12 inches) (5 inches) Substitute. 12 in. = 30 square inches Multiply. YOUR TURN Find the area of the triangle. Personal Math Trainer 3. Online Assessment and Intervention 14 in. 8.5 in. my.hrw.com Math Talk A= Mathematical Processes Why can you also write the formula for the area of a bh triangle as A = __ ? 2 Problem Solving Using Area of Triangles © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©JOHN MACDOUGALL/AFP/Getty Images You can use the formula for the area of a triangle to solve real-world problems. Math On the Spot EXAMPL 2 EXAMPLE 6.8.D Each triangular face of the Pyramid of Peace in Kazakhstan is made up of 25 smaller equilateral triangles. These triangles have measurements as shown in the diagram. What is the area of one of the smaller equilateral triangles? STEP 1 Identify the length of the base and the height of the triangle. my.hrw.com 10.4 m 12 m b = 12 m and h = 10.4 m STEP 2 Use the formula to find the area of the triangle. A = _12 bh = _12 (12) (10.4) Substitute. Multiply. = 62.4 The area of one small equilateral triangle is 62.4 m2. Lesson 16.2 455 YOUR TURN 4. Personal Math Trainer Amy needs to order a shade for a triangular-shaped window that has a base of 6 feet and a height of 4 feet. What is the area of the shade? Online Assessment and Intervention my.hrw.com Guided Practice 1. Show how you can use a copy of the triangle to form a rectangle. Find the area of the triangle and the area of the rectangle. What is the relationship between the areas? (Explore Activities 1 and 2, Example 1) 8 in. 14 in. Triangle: A = _12 bh Rectangle: A = bh = )( ) in2 =( )( = in2 2. A pennant in the shape of a triangle has a base of 12 inches and a height of 30 inches. What is the area of the pennant? (Example 2) A = _12 bh = _12 ( = ? ? )( ) in2 ESSENTIAL QUESTION CHECK-IN 3. How do you find the area of a triangle? 456 Unit 5 # ) '/ 3Ý /9/ 4 % © Houghton Mifflin Harcourt Publishing Company = _12 ( Name Class Date 16.2 Independent Practice Personal Math Trainer 6.8.B, 6.8.D my.hrw.com Online Assessment and Intervention Find the area of each triangle. 4. 5. 20 ft 10 cm 24 ft 15 cm 6. 7. 18 ft 32 ft 17 in. 12 in. 8. What is the area of a triangle that has a base of 15_14 in. and a height of 18 in.? © Houghton Mifflin Harcourt Publishing Company 10. A triangular plot of land has the dimensions shown in the diagram. What is the area of the land? 9. A right triangle has legs that are 11 in. and 13 in. long. What is the area of the triangle? 11. The front part of a tent has the dimensions shown in the diagram. What is the area of this part of the tent? 20 km 5 ft 30 km 12. Multistep The sixth-grade art students are making a mosaic using tiles in the shape of right triangles. Each tile has leg measures of 3 centimeters and 5 centimeters. If there are 200 tiles in the mosaic, what is the area of the mosaic? 8 ft 13. Critique Reasoning Monica has a triangular piece of fabric. The height of the triangle is 15 inches and the triangle’s base is 6 inches. Monica says that the area of 2 the fabric is 90 in . What error did Monica make? Explain your answer. Lesson 16.2 457 14. Show how you can use the given triangle and its two smaller right triangles to form a rectangle. What is the relationship between the area of the original triangle and the area of the rectangle? 12 in. 18 in. FOCUS ON HIGHER ORDER THINKING Work Area 15. Communicate Mathematical Ideas Explain how the areas of a triangle and a parallelogram with the same base and height are related. 17. What If? A right triangle has an area of 18 square inches. a. If the triangle is an isosceles triangle, what are the lengths of the legs of the triangle? b. If the triangle is not an isosceles triangle, what are all the possible lengths of the legs, if the lengths are whole numbers? 458 Unit 5 © Houghton Mifflin Harcourt Publishing Company 16. Analyze Relationships A rectangle and a triangle have the same area. If their bases are the same lengths, how do their heights compare? Justify your answer. LESSON 16.3 ? Solving Area Equations ESSENTIAL QUESTION Expressions, equations, and relationships—6.8.C Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles … where dimensions are positive rational numbers. Also 6.8.D How do you use equations to solve problems about area of rectangles, parallelograms, trapezoids, and triangles? Problem Solving Using the Area of a Triangle Recall that the formula for the area of a triangle is A = _12 bh. You can also use the formula to find missing dimensions if you know the area and one dimension. Math On the Spot my.hrw.com EXAMPL 1 EXAMPLE 6.8.D The Hudson High School wrestling team just won the state tournament and has been awarded a triangular pennant to hang on the wall in the school gymnasium. The base of the pennant is 1.5 feet long. It has an area of 2.25 square feet. What is the height of the pennant? © Houghton Mifflin Harcourt Publishing Company A = _12 bh 1.5 ft Write the formula. 2.25 = _12 (1.5)h Use the formula to write an equation. 2.25 = 0.75h 1 Multiply __ and 1.5. 2 2.25 ____ ____ = 0.75 h 0.75 0.75 Divide both sides of the equation by 0.75. 3= h Math Talk Mathematical Processes How can you use units in the formula to confirm that the units for the height are in feet? The height of the pennant is 3 feet. YOUR TURN 1. Renee is sewing a quilt whose pattern contains right triangles. Each quilt piece has a height of 6 in. and an area of 24 in2. How long is the base of each quilt piece? Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 16.3 459 Writing Equations Using the Area of a Trapezoid You can use the formula for area of a trapezoid to write an equation to solve a problem. Math On the Spot my.hrw.com EXAMPLE 2 6.8.C A garden in the shape of a trapezoid has an area of 44.4 square meters. One base is 4.3 meters and the other base is 10.5 meters long. The height of the trapezoid is the width of the garden. How wide is the garden? A = _12 h (b1 + b2) Math Talk Mathematical Processes How can you check that the answer is reasonable? 4.3 m 10.5 m Write the formula. 44.4 = _12 h (4.3 + 10.5) Use the formula to write an equation. 44.4 = _12 h (14.8) Add inside parentheses. 44.4 = 7.4 h 1 and 14.8. Multiply __ 2 44.4 ___ ____ = 7.4 h 7.4 7.4 Divide both sides of the equation by 7.4. 6= h The garden is 6 meters wide. Reflect Communicate Mathematical Ideas Explain why the first step after substituting is addition. YOUR TURN 3. Personal Math Trainer Online Assessment and Intervention my.hrw.com 460 Unit 5 The cross section of a water bin is shaped like a trapezoid. The bases of the trapezoid are 18 feet and 8 feet long. It has an area of 52 square feet. What is the height of the cross section? © Houghton Mifflin Harcourt Publishing Company 2. Solving Multistep Problems You can write equations to solve real-world problems using relationships in geometry. EXAMPL 3 EXAMPLE Problem Solving 6.8.D Math On the Spot my.hrw.com John and Mary are using rolls of fabric to make a rectangular stage curtain for their class play. The rectangular piece of fabric on each roll measures 2.5 feet by 15 feet. If the area of the curtain is 200 square feet, what is the least number of rolls they need? Analyze Information Rewrite the question as a statement. • Find the least number of rolls of fabric needed to cover an area of 200 ft2. List the important information. • Each roll of fabric is a 2.5 foot by 15 foot rectangle. • The area of the curtain is 200 square feet. Formulate a Plan Write an equation to find the area of each roll of fabric. Use the area of the curtain and the area of each roll to write an equation to find the least number of rolls. Solve STEP 1 Write an equation to find the area of each roll of fabric. A = lw © Houghton Mifflin Harcourt Publishing Company A = 15 · 2.5 A = 37.5 ft2 STEP 2 Write an equation to find the least number of rolls. n = 200 ÷ 37.5 n = 5_13 STEP 3 The problem asks for the least number of rolls needed. Since 5 rolls will not be enough, they will need 6 rolls to make the curtain. John and Mary will need 6 rolls of fabric to make the curtain. Justify and Evaluate The area of each roll is about 38 ft2. Since 38 ft2 · 6 = 228 ft2, the answer is reasonable. Lesson 16.3 461 YOUR TURN Personal Math Trainer 4. Online Assessment and Intervention A parallelogram-shaped field in a park needs sod. The parallelogram has a base of 21.5 meters and a height of 18 meters. The sod is sold in pallets of 50 square meters. How many pallets of sod are needed to fill the field? my.hrw.com Guided Practice 1. A triangular bandana has an area of 70 square inches. The height of the triangle is 8_34 inches. Write and solve an equation to find the length of the base of the triangle. (Example 1) 2. The top of a desk is shaped like a trapezoid. The bases of the trapezoid are 26.5 and 30 centimeters long. The area of the desk is 791 square centimeters. The height of the trapezoid is the width of the desk. Write and solve an equation to find the width of the desk. (Example 2) 3. Taylor wants to paint his rectangular deck that is 42 feet long and 28 feet wide. A gallon of paint covers about 350 square feet. How many gallons of paint will Taylor need to cover the entire deck? (Example 3) Write an equation to find the of the deck. Write an equation to find the . Write and solve the equation. Taylor will need ? ? gallons of paint. ESSENTIAL QUESTION CHECK-IN 4. How do you use equations to solve problems about area of rectangles, parallelograms, trapezoids, and triangles? 462 Unit 5 © Houghton Mifflin Harcourt Publishing Company Write and solve the equation. Name Class Date 16.3 Independent Practice 6.8.C, 6.8.D Personal Math Trainer my.hrw.com Online Assessment and Intervention 5. A window shaped like a parallelogram has an area of 18 _13 square feet. The height of the window is 3 _13 feet. How long is the base of the window? 6. A triangular sail has a base length of 2.5 meters. The area of the sail is 3.75 square meters. How tall is the sail? 7. A section in a stained glass window is shaped like a trapezoid. The top base is 4 centimeters and the bottom base is 2.5 centimeters long. If the area of the section of glass is 3.9 square centimeters, how tall is the section? 8. Multistep Amelia wants to paint three walls in her family room. Two walls are 26 feet long by 9 feet wide. The other wall is 18 feet long by 9 feet wide. a. What is the total area of the walls that Amelia wants to paint? b. Each gallon of paint covers about 250 square feet. How many gallons of paint should Amelia buy to paint the walls? © Houghton Mifflin Harcourt Publishing Company 9. Critical Thinking The area of a triangular block is 64 square inches. If the base of the triangle is twice the height, how long are the base and the height of the triangle? 10. Multistep Alex needs to varnish the top and the bottom of a dozen rectangular wooden planks. The planks are 8 feet long and 3 feet wide. Each pint of varnish covers about 125 square feet and costs $3.50. a. What is the total area that Alex needs to varnish? b. How much will it cost Alex to varnish all the wooden planks? 11. Multistep Leia cuts congruent triangular patches with an area of 45 square centimeters from a rectangular piece of fabric that is 18 centimeters long and 10 centimeters wide. How many of the patches can Leia cut from 32 pieces of the fabric? 12. Multistep A farmer needs to buy fertilizer for two fields. One field is shaped like a trapezoid, and the other is shaped like a triangle. The trapezoidal field has bases that are 35 and 48 yards and a height of 26 yards. The triangular field has the same height and a base of 39 yards. Each bag of fertilizer covers 150 square yards. Use a problem solving model to find how many bags of fertilizer the farmer needs to buy. Lesson 16.3 463 13. A tennis court for singles play is 78 feet long and 27 feet wide. a. The court for doubles play has the same length but is 9 feet wider than the court for singles play. How much more area is covered by the tennis court used for doubles play? b. The junior court for players 8 and under is 36 feet long and 18 feet wide. How much more area is covered by the tennis court used for singles play than by the junior court? c. The court for players 10 and under has the same width but is 18 feet shorter than the court for singles play. How much more area is covered by the tennis court used for singles play? 14. Draw Conclusions The cross section of a metal ingot is a trapezoid. The cross section has an area of 39 square centimeters. The top base of the cross section is 12 centimeters. The length of the bottom base is 2 centimeters greater than the top base. How tall is the metal ingot? Explain. Work Area FOCUS ON HIGHER ORDER THINKING 15. Analyze Relationships A mirror is made of two congruent parallelograms as shown in the diagram. The parallelograms have a combined area of 9 _13 square yards. The height of each parallelogram is 1 _13 yards. b. What is the area of the smallest rectangle of wall that the mirror could fit on? 16. Persevere in Problem Solving A watercolor painting is 20 inches long by 9 inches wide. Ramon makes a mat that adds a 1-inch-wide border around the painting. What is the area of the mat? 464 Unit 5 1 in. 20 in. 9 in. © Houghton Mifflin Harcourt Publishing Company 1 yd — 2 a. How long is the base of each parallelogram? LESSON 16.4 ? Solving Volume Equations ESSENTIAL QUESTION Equations, expressions, and relationships— 6.8.C Write equations that represent problems related to . . . volume of right rectangular prisms where dimensions are positive rational numbers. Also 6.8.D How do you write equations to solve problems involving volume of right rectangular prisms? Problem Solving by Finding Volume To find the volume of a box, which is in the shape of a rectangular prism, you can multiply the length, the width, and the height. The volume of a threedimensional shape is always in cubic units, such as cubic meters or cubic inches. Math On the Spot my.hrw.com Volume of a Rectangular Prism The volume V of a rectangular prism is the product of its length ℓ, its width w, and its height h. h V = ℓwh Math Talk ℓ Mathematical Processes w Why are V = ℓwh and V = Bh both formulas for the volume of a rectangular prism? © Houghton Mifflin Harcourt Publishing Company EXAMPL 1 EXAMPLE 6.8.D A rectangular swimming pool is 25 meters long and 17 _12 meters wide. It has an average depth of 1_12 meters. What is the volume of the pool? 1 12 m 17 12 m Label the rectangular prism to represent the pool. ℓ = 25 meters w = 17 _12 meters 25 m h = 1 _12 meters Use the formula to write an equation. V = ℓwh V = 25 · 17 _12 · 1 _12 35 _ ·3 = 25 · __ 2 2 2,625 Write mixed numbers as fractions greater than 1. = ____ 4 Multiply. = 656 _14 cubic meters Write as a mixed number in simplest form. Lesson 16.4 465 YOUR TURN Personal Math Trainer Online Assessment and Intervention 1. Miguel has a toolbox that measures 18 _12 inches by 12 _12 inches by 4 inches. What is the volume of the toolbox? V= cubic inches my.hrw.com Writing Equations Using the Volume of a Rectangular Prism Math On the Spot You can use the formula for the volume of a rectangular prism to write an equation. Then solve the equation to find missing measurements for a prism. my.hrw.com EXAMPLE 2 6.8.C Samuel has an ant farm with a volume of 375 cubic inches. The width of the ant farm is 2.5 inches and the length is 15 inches. What is the height of Samuel’s ant farm? V = ℓwh Write the formula. 375 = 15 · 2.5 · h Use the formula to write an equation. Multiply. 375 37.5h ____ = ____ 37.5 37.5 Divide both sides of the equation by 37.5. 10 = h The height of the ant farm is 10 inches. Reflect 2. Communicate Mathematical Ideas Explain how you would find the solution to Example 2 using the formula V = Bh. YOUR TURN Personal Math Trainer Online Assessment and Intervention my.hrw.com 466 Unit 5 ? 3. Find the height of this shape, which has a 15 volume of __ cubic feet. 16 3 4 ft 1 2 ft © Houghton Mifflin Harcourt Publishing Company • Image Credits: Thom Lang/CORBIS 375 = 37.5h Solving Multistep Problems One cubic foot of water equals approximately 7.5 gallons and weighs approximately 62.43 pounds. EXAMPL 3 EXAMPLE 6.8.D Math On the Spot my.hrw.com The classroom aquarium holds 30 gallons of water. It is 0.8 feet wide and has a height of 2 feet. Find the length of the aquarium. Divide the total number STEP 1 Find the volume of the classroom of gallons by the unit aquarium in cubic feet. rate to find the number 30 gallons of cubic feet. __________________ = 4 cubic feet 7.5 gallons per cubic foot The volume of the classroom aquarium is 4 cubic feet. STEP 2 Find the length of the aquarium. V = ℓwh Write the formula for volume. 4 = ℓ · 0.8 · 2 Use the formula to write an equation. 4 = ℓ(1.6) Multiply. ℓ(1.6) 4 ___ = ____ 1.6 1.6 Divide both sides of the equation by 1.6. 2.5 = ℓ The length of the classroom aquarium is 2.5 feet. Reflect © Houghton Mifflin Harcourt Publishing Company 4. Persevere in Problem Solving How much does the water in the classroom aquarium weigh? Explain. YOUR TURN 5. An aquarium holds 33.75 gallons of water. It has a length of 2 feet and a height of 1.5 feet. What is the volume of the aquarium? What is the width of the aquarium? Explain. Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 16.4 467 Guided Practice 1. Find the volume of this rectangular prism. (Example 1) 1 1 2 in. V = ℓwh V = 3_12 · · V = _____ · ______ · _____ · ______ 2 2 1 4 1 3 2 in. 3 in. V= The volume of the rectangular prism is cubic inches. 2. Write an equation to find the width of the rectangular prism. Show your work. (Example 2) V = 6,336 cm3 18 cm ? cm 16 cm ? ? ESSENTIAL QUESTION CHECK-IN 4. How do you solve problems about volume of right rectangular prisms? 468 Unit 5 © Houghton Mifflin Harcourt Publishing Company 3. One red clay brick weighs 5.76 pounds. The brick is 8 inches long and 2_14 inches wide. If the clay weighs 0.08 pounds per cubic inch, what is the volume of the brick? Write an equation to find the height of the brick. Show your work. (Example 3) Name Class Date 16.4 Independent Practice Personal Math Trainer 6.8.C, 6.8.D my.hrw.com 5. Jala has an aquarium in the shape of a rectangular prism with the dimensions shown. What is the height of the aquarium? Height = 6. Find the volume of a juice box that is 3 in. by 1_21 in. by 4 in. Online Assessment and Intervention ? V = 3,758.75 cubic inches 12.5 in. 24.25 in. Volume = 7. Find the width of a cereal box that has a volume of 3,600 cm3 and is 20 cm long and 30 cm high. Width = 8. Bill has a box of markers that has a base of 8 cm by 20 cm and a height of 6 cm. Martin’s pencil box has a height of 4 cm and a base that is 15 cm by 16 cm. Bill says his marker box has the same volume as Martin’s pencil box. Is Bill right? Explain. © Houghton Mifflin Harcourt Publishing Company 9. Physical Science A small bar of gold measures 40 mm by 25 mm by 2 mm. One cubic millimeter of gold weighs about 0.0005 ounces. Find the volume in cubic millimeters and the weight in ounces of this small bar of gold. 10. History The average stone on the lowest level of the Great Pyramid in Egypt was a rectangular prism 5 feet long by 5 feet high by 6 feet deep and weighed 15 tons. What was the volume of the average stone? How much did one cubic foot of this stone weigh? 11. A freshwater fish is healthiest when there is at least one gallon of water for every inch of its body length. Roshel wants to put a goldfish that is about 2_12 inches long in her tank. Roshel’s tank is 7 inches long, 5 inches wide, and 7 inches high. The volume of 1 gallon of water is about 231 cubic inches. a. How many gallons of water would Roshel need for the fish? b. What is the volume of Roshel’s tank? c. Is her fish tank large enough for the fish? Explain. Lesson 16.4 469 12. A box of crackers is a rectangular box with the dimensions shown. The box is one-fourth full. What is the volume of 4 in. crackers in the box? 4 in. FOCUS ON HIGHER ORDER THINKING 8 in. Work Area 13. Multistep Larry has a clay brick that is 7 inches long, 3.5 inches wide, and 1.75 inches thick, the same size as the gold stored in Ft. Knox in the form of gold bars. Find the volume of this brick. If the weight of the clay in the brick is 0.1 pound per cubic inch and the weight of the gold is 0.7 pounds per cubic inch, find the weight of the brick and the gold bar. Round all answers the nearest tenth. Volume of the brick or bar = Weight of the brick = Weight of the gold bar = cubic inches pounds pounds 15. Multiple Representations Use the formula V = Bh to write a different version of this formula that you could use to find the area of the base B of a rectangular prism if you know the height h and the volume V. Explain what you did to find this equation. 16. Communicate Mathematical Ideas Explain how you could find the volume of a cube that has an edge of e. 17. Justify Reasoning Mariel says that a jewelry box that is 3 inches high, 4_12 inches wide, and 5 inches long has a volume of 67_12 inches. Katy says that answer is not quite correct. What is the error in Mariel’s answer? 470 Unit 5 © Houghton Mifflin Harcourt Publishing Company 14. Represent Real-World Problems Luisa’s toaster oven, which is in the shape of a rectangular prism, has a base that is 55 cm long by 40 cm wide. It is 30 cm high. Luisa wants to buy a different oven with the same volume but a smaller length, so it will fit better on her kitchen counter. What is a possible set of dimensions for this different oven? MODULE QUIZ Ready Personal Math Trainer 16.1 Area of Quadrilaterals 1. Find the area of the figure. Online Assessment and Intervention my.hrw.com 1 yd 12 — 2 1 yd 17 — 5 16.2 Area of Triangles 2. Find the area of the triangle. 16.3 Solving Area Equations 14 ft 17 ft 3. A triangular pane of glass has a height of 30 inches and an area of 270 square inches. What is the length of the base of the pane? 4. A tabletop in the shape of a trapezoid has an area of 6,550 square centimeters. Its longer base measures 115 centimeters, and the shorter base is 85 centimeters. What is the height? © Houghton Mifflin Harcourt Publishing Company 16.4 Solving Volume Equations 5. A rectangular shoebox has a volume of 728 cubic inches. The base of the shoebox measures 8 inches by 6.5 inches. How long is the shoebox? ESSENTIAL QUESTION 6. How can you use equations to solve problems involving area and volume? Module 16 471 Personal Math Trainer MODULE 16 MIXED REVIEW Texas Test Prep Selected Response 1. What is the area of the rhombus shown below? my.hrw.com Online Assessment and Intervention 4. The trapezoid below has an area of 1,575 cm2. 63 cm 23 in. 28 in. 27 cm Which equation could you solve to find the height of the trapezoid? A 161 in2 C B 322 in2 D 966 in2 644 in2 2. What is the area of the triangle shown below? A 45h = 1,575 C 850.5h = 1,575 B 90h = 1,575 D 1,701h = 1,575 Gridded Response 5. Cindy is designing a rectangular fountain in a courtyard. The rest of the courtyard will be covered in stone. 22 ft Courtyard 6 ft 3.7 mm 12 ft Fountain A 4.44 mm2 B 5.92 mm C 2 8.88 mm2 The part of the courtyard that will be covered in stone has an area of 246 ft2. What is the width of the fountain in feet? D 17.76 mm2 3. A rectangular prism has a volume of 912 cubic meters. It has a length of 19 meters and a width of 12 meters. Which equation could be solved to find the height of the rectangular prism? 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 A 114h = 912 5 5 5 5 5 5 B 228h = 912 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 C 15.5h = 912 D 31h = 912 472 . Unit 5 © Houghton Mifflin Harcourt Publishing Company 4.8 mm UNIT 5 Study Guide MODULE MODULE ? 15 15 Review Angles, Triangles, and Equations Key Vocabulary parallelogram (paralelogramo) rhombus (rombo) trapezoid (trapecio) ESSENTIAL QUESTION How can you use angles, triangles, and equations to solve real-world problems? EXAMPLE 1 Find the missing angle measure in each triangle. A B 51° y 21° 18° 51 + 90 + x = 180 18 + 21 + y = 180 141 + x = 180 39 + y = 180 x x = 39° y = 141° EXAMPLE 2 The triangle shown has approximate side lengths of 5 cm, 5.8 cm, and 3 cm. Match each side with its correct length. A © Houghton Mifflin Harcourt Publishing Company 31° AB = 5 cm 59° C BC = 3 cm The shortest side is opposite the smallest angle. AC = 5.8 cm The longest side is opposite the largest angle. B EXERCISES Tell whether a triangle can have sides with the given lengths. If it cannot, give an inequality that shows why not. (Lesson 15.1) 1. 5 in., 12 in., 13 in. 2. 4.5 ft, 5.5 ft, 11 ft Unit 5 473 Find each missing angle measure. Classify each triangle as acute, obtuse, or right. (Lesson 15.2) 3. 4. 82° 41° x 38° 49° x Match each side length with its correct measure. Classify each triangle as scalene, isosceles, or equilateral. (Lesson 15.3) 5. The side lengths of triangle ABC are 6.4 ft, 10 ft, and 6.4 ft. A AB = 102° BC = 39° AC = 39° B C 6. The side length of ZX is 17 cm. X XY = 60° YZ = 60° 60° Z ? 16 16 Area and Volume Equations ESSENTIAL QUESTION How can you use area and volume equations to solve real-world problems? 4 in. EXAMPLE 1 Find the area of the trapezoid. A = _12 (h) (b1 + b2) 10 in. A = _12 (10) (7 + 4) A = 55 in2 474 Unit 5 7 in. Key Vocabulary parallelogram (paralelogramo) rhombus (rombo) trapezoid (trapecio) © Houghton Mifflin Harcourt Publishing Company MODULE MODULE Y EXAMPLE 2 A triangular sail for a sailboat has a height of 30 feet and an area of 330 square feet. Find the base length of the sail. A = _12 bh () 330 = _12 30b b = 22 ft EXAMPLE 3 A cubic centimeter of gold weighs approximately 19.32 grams. Find the weight of a brick of gold that has a height of 6 centimeters, width of 3 centimeters, and length of 8 centimeters. V = lwh 8 cm V = 8 (3) (6) V = 144 cm3 6 cm 3 cm The weight of the gold is 144 × 19.32 grams, which is 2,782.08 grams. EXERCISES Find the area of each figure. (Lessons 16.1, 16.2) 24 in. 1. 2. 12 in. 8 ft 8 ft Find the missing measurement. (Lesson 16.3) © Houghton Mifflin Harcourt Publishing Company 3. 11 m h b 4. 4 mm A = 62.5 m2 A = 26 mm2 14 m Find the volume of each rectangular prism. (Lesson 16.4) 5. 6. A rectangular prism with a width of 7 units, a length of 8 units, and a height 6 in. 8 in. 20 in. of 2 units Unit 5 475 7. Jelani is ordering a piece of glass in the shape of a trapezoid to create a patio table top. Each square foot of glass costs $25. The trapezoid has base lengths of 5 feet and 3 feet and a height of 4 feet. Find the cost of the glass. (Lesson 16.1) Unit 5 Performance Tasks 1. Theater Set Construction Ahmed and Karina are building scenery of the Egyptian pyramids 1 ft out of plywood for a community play. The pyramids are represented by triangles on a rectangular base. The diagram shows the measurements of the piece of scenery. CAREERS IN MATH 1.5 ft 0.75 ft 1.5 ft 2 ft 1 ft 0.75 ft 5 ft a. They have one sheet of plywood, 3 ft by 6 ft. Will they be able to make the piece using this one sheet? Explain. b. How many square feet of plywood is in the completed piece? Show your work. 2. Cassandra is making a design for a logo. One part of the design is a triangle with two congruent sides. She must draw the triangle with at least one side with length 6 centimeters, and at least one side with length 4 centimeters. Sketch two possible figures that Cassandra could use. Label the side lengths in both figures. 476 Unit 5 © Houghton Mifflin Harcourt Publishing Company c. The pyramids (the triangles) will be painted gray, and the base (the rectangle) will be painted black. How much of each paint color will they use, if one quart covers 45 square feet? Only one side of the model needs to be painted, but two coats of paint will be needed. Show your work. Round to the nearest hundredth of a square foot. Personal Math Trainer UNIT 5 MIXED REVIEW Texas Test Prep Selected Response 1. Part of a large wooden art project will be a triangle formed by joining three boards together. The artist has four boards that measure 16 feet, 11 feet, 7 feet, and 3 feet. Which board could not be used with two of the others to form a triangle? my.hrw.com Online Assessment and Intervention 4. What is the area of the triangle shown below? 17 in. 26 in. A the 3-foot board A 110.5 square inches B the 7-foot board B 221 square inches C the 11-foot board C 442 square inches D the 16-foot board 2. Which of these could be the value of x in the triangle below? A D 884 square inches 5. The trapezoid below has an area of 475 square meters. 28 m 53° 38 © Houghton Mifflin Harcourt Publishing Company 18 m 85° 42° C 26 B 5x A 5 C 8 B 6 D 10 3. What is the area of a trapezoid that has bases measuring 19 centimeters and 23 centimeters, and a height of 14 centimeters? A 105 square centimeters B 266 square centimeters C 294 square centimeters D 322 square centimeters Which equation could you solve to find the height of the trapezoid? A 23h = 475 B 252h = 475 C 46h = 475 D 504h = 475 6. A rectangular prism has a volume of 1,500 cubic centimeters. It has a length of 34 centimeters and a width of 22 centimeters. Which equation could be solved to find the height of the rectangular prism? A 374h = 1,500 B 28h = 1,500 C 748h = 1,500 D 56h = 1,500 Unit 5 477 7. Which expression represents the sum of 59 and x? A 59 + x B 59 ÷ x Hot ! Tip It is helpful to draw or redraw a figure. Answers to geometry problems may become clearer as you redraw the figure. 11. What is the measure, in degrees, of the missing angle in the triangle below? C 59x D 59 - x Q 8. Which number has more than two factors? 39° A 19 B 23 C 25 D 29 66° S 9. Which of the following statements about rational numbers is not correct? numbers. B All integers are also rational numbers. C All rational numbers can be written in the form _a where b ≠ 0. b D Rational numbers cannot be negative. Gridded Response 10. What is the measure of the missing angle in a triangle that contains angle measures of 37° and 59°? . 478 Unit 5 . 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 12. Janice wants to buy carpet for a trapezoidshaped room. The bases of the trapezoid are 12 feet and 14 feet, and the height is 15 feet. If the carpet she likes is $5.50 per square foot, how much will new carpet for the room cost in dollars? 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 0 0 0 0 0 0 5 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 2 2 2 2 2 2 7 7 7 7 7 7 3 3 3 3 3 3 8 8 8 8 8 8 4 4 4 4 4 4 9 9 9 9 9 9 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 . © Houghton Mifflin Harcourt Publishing Company A All whole numbers are also rational R