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Two-Dimensional Figures • Lesson 10-1 Identify the relationships of parallel and intersecting lines. • Lesson 10-2 triangles. Identify properties of congruent • Lesson 10-3 Identify and draw transformations. • Lessons 10-4 and 10-6 Classify and find angle measures of polygons. • Lessons 10-5, 10-7, and 10-8 Find the area of polygons and irregular figures, and find the area and circumference of circles. Two-dimensional shapes are usually found in many real-world objects. The properties of two-dimensional figures can be used to solve real-world problems dealing with snowflakes. For example, if you know the exact shape of a snowflake, you can find the measure of an interior angle in a snowflake. You will solve a problem about snowflakes in Lesson 10-6. 490 Chapter 10 Two-Dimensional Figures Key Vocabulary • • • • • parallel lines (p. 492) transformation (p. 506) quadrilateral (p. 513) polygon (p. 527) circumference (p. 533) Prerequisite Skills To be successful in this chapter, you'll you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter X. 10. For Lessons 10-1 and 10-4 Solve Equations Solve each equation. (For review, see Lessons 3-3 and 3-5.) 1. x 46 90 2. x 35 180 3. 2x 12 90 4. 3x 24 180 5. 2x 34 90 6. 4x 44 180 7. 5x 165 360 8. 4x 184 360 For Lessons 10-5, 10-7, and 10-8 Multiply Decimals Find each product. Round to the nearest tenth, if necessary. (For review, see page 715.) 9. (5.5)(8) 1 13. (4.3)(5.8) 2 10. (7.5)(3.4) 11. (6.3)(11.4) 1 12. (8)(2.5) 14. (3.14)(7) 15. (2)(3.14)(1.7) 16. 2(3.1)(3.14) 2 For Lesson 10-5 Add Mixed Numbers Find each sum. (For review, see Lesson 5-7.) 1 2 2 3 5 3 21. 3 1 8 4 17. 5 4 1 3 3 4 7 3 22. 2 4 10 5 18. 2 3 3 1 8 2 2 5 23. 2 3 3 9 1 5 4 6 2 4 24. 5 3 3 5 19. 1 2 20. 6 1 Polygons Make this Foldable to help you organize your notes. Begin with four plain sheets of 8 12" by 11" paper, eight index cards, and glue. Fold Fold a sheet of paper in half widthwise. Repeat Steps 1 and 2 Repeat three times. Then glue all four pieces together to form a booklet. Open and Fold Again Open and fold the bottom to form a pocket. Glue edges. Label Label each pocket. Place an index card in each pocket M es Trapeoznkeylike ngl Tria oids Reading and Writing As you read and study the chapter, write the name of a two-dimensional figure on each index card, draw a diagram, and write a definition or describe the characteristics of each figure. Chapter 10 Two-Dimensional Figures 491 Line and Angle Relationships • Identify the relationships of angles formed by two parallel lines and a transversal. Vocabulary • • • • • • • • • • • • parallel lines transversal interior angles exterior angles alternate interior angles alternate exterior angles corresponding angles vertical angles adjacent angles complementary angles supplementary angles perpendicular lines Study Tip Parallel Lines Arrowheads are often used in figures to indicate parallel lines. • Identify the relationships of vertical, adjacent, complementary, and supplementary angles. are parallel lines and angles related? Let’s investigate what happens when two horizontal lines are intersected by a third line. a. Trace two of the horizontal lines on a sheet of notebook paper. Then draw another line that intersects the 1 2 horizontal lines. 4 3 b. Label the angles as shown. 5 6 8 7 c. Find the measure of each angle. d. What do you notice about the measures of the angles? e. Which angles have the same measure? f. What do you notice about the measures of the angles that share a side? PARALLEL LINES AND A TRANSVERSAL In geometry, two lines in a plane that never intersect are parallel lines . Lines m and n are parallel. Using symbols, m n. m Parallel lines have no point of intersection. n When two parallel lines are intersected by a third line called a transversal , eight angles are formed. Names of Special Angles The eight angles formed by parallel lines and a transversal have special names. • Interior angles lie inside the parallel lines. 3, 4, 5, 6 • Exterior angles lie outside the parallel lines. transversal 1, 2, 7, 8 • Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. 3 and 5, 4 and 6 • Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. 1 and 7, 2 and 8 • Corresponding angles are in the same position on the parallel lines in relation to the transversal. 1 and 5, 2 and 6, 3 and 7, 4 and 8 492 Chapter 10 Two-Dimensional Figures 1 2 4 3 5 6 8 7 In Lesson 9-4, you learned that line segments are congruent if they have the same measure. Similarly, angles are congruent if they have the same measure. Parallel Lines Cut by a Transversal If two parallel lines are cut by a transversal, then the following pairs of angles are congruent. • Corresponding angles are congruent. • Alternate interior angles are congruent. • Alternate exterior angles are congruent. Concept Check How many angles are formed when two parallel lines are intersected by a transversal? Example 1 Find Measures of Angles m In the figure at the right, m n and t is a transversal. If m1 68°, find m5 and m6. 1 3 7 4 Since 1 and 5 are corresponding angles, they are congruent. So, m5 68°. n 5 2 8 6 Since 1 and 6 are alternate exterior angles, they are congruent. So m6 68°. t INTERSECTING LINES AND ANGLES Reading Math Congruent Angles Angle 1 is congruent to angle 2. This is written 1 2. The measure of 1 is equal to the measure of 2. This is written m1 m2. Other pairs of angles have special relationships. When two lines intersect, they form two pairs of opposite angles called vertical angles . Vertical angles are congruent. The symbol for is congruent to is . 1 and 2 are vertical angles. 1 2 3 and 4 are vertical angles. 3 4 3 1 2 4 When two angles have the same vertex, share a common side, and do not overlap, they are adjacent angles . 1 and 2 are adjacent angles. mAOB m1 m2 A 2 1 B O If the sum of the measures of two angles is 90°, the angles are complementary . 4 1 2 m1 50°, m2 40° m1 m2 90° www.pre-alg.com/extra_examples 3 m3 60°, m4 30° m3 m4 90° Lesson 10-1 Line and Angle Relationships 493 If the sum of the measures of two angles is 180°, the angles are supplementary . 1 2 3 m1 140°, m2 40° m1 m2 180° 4 m3 45°, m4 135° m3 m4 180° Lines that intersect to form a right angle are perpendicular lines . right angle Standardized Example 2 Find a Missing Angle Measure Test Practice Multiple-Choice Test Item If mB 87° and A and B are supplementary, what is mA? A 3° B 87° C Test-Taking Tip It is a good idea to review definitions of key terms, such as supplementary, before taking the test. 93° D 87˚ 95° A B Read the Test Item Since A and B are supplementary, mA mB 180°. Solve the Test Item mA mB 180° Supplementary angles mA 87° 180° Replace mB with 87°. mA 87° 87° 180° 87° Subtract 87 from each side. mA 93° The answer is C. Example 3 Find Measures of Angles Study Tip Checking Reasonableness of Results To check your answer, add to see if the sum of the measures of the angles is 90. Since 54 36 = 90, the answer is correct. ALGEBRA Angles ABC and FGH are complementary. If mABC x 8 and mFGH x 10, find the measure of each angle. Step 1 Find the value of x. mABC mFGH 90° Complementary angles (x 8) (x 10) 90° Substitution 2x 2 90° Combine like terms. 2x 92° Add 2 to each side. x 46° Divide each side by 2. Step 2 Replace x with 46 to find the measure of each angle. mABC x 8 46 8 or 54 mFGH x 10 46 10 or 36 So, mABC 54° and mFGH 36°. 494 Chapter 10 Two-Dimensional Figures Example 4 Apply Angle Relationships SAFETY A lifeguard chair is shown. If m1 105°, find m4 and m6. Since 1 and 4 are vertical angles, they are congruent. So, m4 105°. 1 6 7 4 3 8 5 2 Since 6 and 1 are supplementary, the sum of their measures is 180°. 180 105 75. So, m6 75°. Line and Angle Relationships Parallel Lines Perpendicular Lines Vertical Angles n mn ab a 1 m 2 4 1 ≅ 3 2 ≅ 4 3 b Adjacent Angles Supplementary Angles C D 1 1 2 A B mABC m1 m2 Concept Check Complementary Angles 2 1 m1 m2 90° 2 m1 m2 180° 1. Explain the difference between complementary and supplementary angles. 2. OPEN ENDED Draw a pair of adjacent, supplementary angles. Guided Practice In the figure at the right, m and k is a transversal. If m1 56°, find the measure of each angle. 3. 2 4. 3 5. 4 Find the value of x in each figure. 6. x˚ 140˚ ᐉ k 1 2 3 7. 152˚ m 4 x˚ 8. ALGEBRA If mN 3x and mM 2x and M and N are supplementary, what is the measure of each angle? Standardized Test Practice 9. If mB 26° and A and B are complementary, what is mA? A 154° B 90° C 26° D 64° Lesson 10-1 Line and Angle Relationships 495 Practice and Apply g In the figure at the right, g h and t is a transversal. If m4 53°, find the measure of each angle. 10. 1 11. 5 For Exercises See Examples 10–15, 26–28 16–21 22–25, 29–32 1, 4 12. 7 13. 8 2 3 14. 2 15. 3 4 1 3 2 5 h 7 8 6 t Extra Practice See page 747. Find the value of x in each figure. 16. 17. 45˚ x˚ 18. 148˚ x˚ 31˚ x˚ 19. x˚ 20. 5˚ 21. 8x ˚ 4x ˚ 5x ˚ 5x ˚ 22. Find mA if mB 17° and A and B are complementary. 23. Angles P and Q are supplementary. Find mP if mQ 139°. 24. ALGEBRA Angles J and K are complementary. If mJ x 9 and mK x 5, what is the measure of each angle? 25. ALGEBRA Find mE if E and F are supplementary, mE 2x 15, and mF 5x 38. 26. SAFETY Refer to Example 4 on page 495. Find the measure of angles 2, 3, 5, 7, and 8. CONSTRUCTION For Exercises 27 and 28, use the following information and the diagram shown. To measure the angle between a sloped ceiling and a wall, a carpenter uses a plumb line (a string with a weight attached). 27. If mYXB 68°, what is mXBC? A X B Y C 28. What type of angles are YXB and XBC? ALGEBRA In the figure at the right, m and t is a transversal. Find the value of x for each of the following. 29. m2 2x 3 and m4 4x 7 30. m8 4x 32 and m5 5x 50 t ᐉ 1 4 m 5 8 2 3 6 7 31. m7 10x 15 and m3 7x 42 32. ALGEBRA The measure of the supplement of an angle is 15° less than four times the measure of the complement. Find the measure of the angle. 496 Chapter 10 Two-Dimensional Figures 33. CRITICAL THINKING Suppose two parallel lines are cut by a transversal. How are the interior angles on the same side of the transversal related? Answer the question that was posed at the beginning of the lesson. How are parallel lines and angles related? Include the following in your answer: • a drawing of parallel lines intersected by a transversal, and • a list of the congruent and supplementary angles. 34. WRITING IN MATH Standardized Test Practice For Exercises 35 and 36, use the diagram. 35. The upper rail is parallel to the lower rail. What is the measure of the angle formed by the upper rail and the first vertical post? A 135° B 100° C 90° D upper rail first x˚ vertical post lower rail 45˚ 45˚ 45° 36. What is the measure of the angle formed by the second vertical post and the lower rail? A 100° B 135° C 45° D 90° Extending the Lesson For Exercises 37–39, use the pairs of graphs shown at the right. 37. How are each pair of graphs related? 38. What seems to be true about the slopes of the graphs? y y y 3x 2 y 2x 3 x O x O y 13 x 1 y 12 x 1 39. Make a conjecture about the slopes of the graphs of perpendicular lines. Maintain Your Skills Mixed Review Use a calculator to find each value to the nearest ten thousandth. (Lesson 9-8) 40. cos 21° 41. sin 63° 42. tan 38° 43. If ABC ∼ DEF, what is the value of x? (Lesson 9-7) B E 10 in. 15 in. Simplify each expression. (Lesson 3-2) 44. 6a (18)a 45. 5m (4)m Getting Ready for the Next Lesson A x in. C D 5 in. F PREREQUISITE SKILL Use a protractor to draw an angle having each measurement. (To review angles, see Lesson 9-3.) 46. 20° 47. 45° 48. 65° 49. 145° 50. 170° www.pre-alg.com/self_check_quiz Lesson 10-1 Line and Angle Relationships 497 A Follow-Up of Lesson 10-1 Constructions Activity 1 Construct a line segment congruent to a given line segment. Draw A B . Then use a straightedge to draw GH so it is longer than A B . Place the tip of the compass at A and the pencil tip at B. Using this setting, place the tip at G. Draw an arc to intersect . Label the intersection J. GH J A B. G B A B A H G J H G Activity 2 Construct an angle congruent to a given angle. Draw DEF. Then use a . straightedge to draw JK D Place the tip of the compass at E. Draw an arc to intersect both sides of DEF to locate points X and Y. Using this setting, place the compass at point J. Draw an . Label the arc to intersect JK intersection A. D X F E K J Place the point of the compass on X and adjust so that the pencil tip is on Y. D F F Using this setting, place the compass at A and draw an arc to intersect the arc drawn in Step 3. Label the intersection M. . MJK ≅ DEF. Draw JM J A M K J Model and Analyze 1. Draw a line segment. Construct a line segment congruent to the one drawn. 2. Draw an angle. Construct an angle congruent to the one drawn. 498 Chapter 10 Two-Dimensional Figures K Step 6 M Y A J Step 5 X E Y E A K Activity 3 Construct the perpendicular bisector of a line segment. Draw X Y . Then place the compass at point X. Use a setting Y. Draw greater than one half of X an arc above and below X Y . Using this setting, place the compass at point Y. Draw an arc above and below XY as shown. Y X M X Y Y X Activity 4 Construct the bisector of an angle. Draw MNP. Then place the compass at point N and draw an arc that intersects both sides of the angle. Label the intersections X and Y. With the compass at point X, draw an arc in the interior of N. Using this setting, place the compass at point Y. Draw another arc. Label the intersection Q. X Y . NQ is the bisector of Draw NQ MNP. M M N Use a straightedge to align the two intersections. Draw a segment that intersects X Y . Label the intersection M. N X Q X P M Y P Y N Q P Model and Analyze 3. In Activity 3, use a ruler to measure X M and M Y . This construction bisects a segment. What do you think bisects means? 4. Draw a line segment. Construct the perpendicular bisector of the segment. 5. In Activity 4, what is true about the measures of MNQ and QNP? is the bisector of MNP? 6. Why do we say NQ Algebra Activity Constructions 499 Congruent Triangles • Identify congruent triangles and corresponding parts of congruent triangles. Vocabulary • congruent • corresponding parts Where are congruent triangles present in nature? Ivy is a type of climbing plant. Most ivy leaves have five major veins. In the photo shown, the veins outlined form two triangles. a. Trace the triangles shown at the right onto a sheet of paper. Then label the triangles. b. Measure and then compare the lengths of the sides of the triangles. c. Measure the angles of each triangle. A How do the angles compare? d. Make a conjecture about the triangles. Reading Math Corresponding Everyday Meaning: matching Math Meaning: having the same position C E B F D CONGRUENT TRIANGLES Figures that have the same size and shape are congruent. The parts of congruent triangles that “match” are corresponding parts. Corresponding Parts of Congruent Triangles • Words If two triangles are congruent, their corresponding sides are congruent and their corresponding angles are congruent. • Model Y Q Slash marks are used to indicate which sides are congruent. X Z Arcs are used to indicate which angles are congruent. P R • Symbols Congruent Angles: X P, Y Q, Z R Y PQ YZ QR PR Congruent Sides: X , , X Z Concept Check 500 Chapter 10 Two-Dimensional Figures What is true about the corresponding angles and sides of congruent triangles? When writing a congruence statement, the letters must be written so that corresponding vertices appear in the same order. For example, for the diagram below, write FGH JKM. G Study Tip FGH JKM K Congruence Statements You can also write a congruence statement as GHF KMJ, HFG MJK, FHG JMK, GFH KJM, and HGF MKJ. F H J Vertex F corresponds to vertex J. Vertex G corresponds to vertex K. Vertex H corresponds to vertex M. M Example 1 Name Corresponding Parts Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement. A Corresponding Angles A Z, B Y, C X Z B C X ABC Corresponding Sides A B Z Y , B C Y X , C A X Z Y ? One congruence statement is ABC ZYX. Congruence statements can be used to identify corresponding parts of congruent triangles. Example 2 Use Congruence Statements If SRT WVU, complete each congruence statement. S ? U ? R ? W V R T ? ? S T ? Explore You know the congruence statement. You need to find the corresponding parts. Plan Use the order of the vertices in SRT WVU to identify the corresponding parts. Solve SRT WVU S corresponds to W so S W. R corresponds to V so R V. T corresponds to U so T U. S corresponds to W, and R corresponds to V so S R W V . R corresponds to V, and T corresponds to U so R T V U . S corresponds to W, and T corresponds to U so S T W U . Examine Draw the triangles, using arcs and slash marks to show the congruent angles and sides. www.pre-alg.com/extra_examples R S U T W V Lesson 10-2 Congruent Triangles 501 You can use corresponding parts to find the measures of angles and sides in a figure that is congruent to a figure with known measures. Example 3 Find Missing Measures LANDSCAPING A brace is used to support a tree and help it to grow straight. In the figure, TRS ERS. a. At what angle is the brace placed against the ground? E and T are corresponding angles. So, they are congruent. Since mT 65°, mE 65°. R brace 8 ft The brace is placed at a 65° angle with the ground. 65˚ b. What is the length of the brace? R RT RE E corresponds to . So, and R T are congruent. Since RT 8 feet, RE 8 feet. T E S The length of the brace is 8 feet. Concept Check 1. Explain when two figures are congruent. 2. OPEN ENDED Draw and label a pair of congruent triangles. Be sure to mark the corresponding parts. Guided Practice For each pair of congruent triangles, name the corresponding parts. Then complete the congruence statement. C K 3. 4. A G C B J KMJ M B E D ? CBE ? Complete each congruence statement if DKJ NAM. 5. J ? 6. A ? 7. JD ? Application 9. TOWERS A tower that supports high voltage power lines is shown at the right. In the tower, ETC YRC. What is the length of RC if EC 10 feet and TC 15 feet? 8. AN E Y C T 502 Chapter 10 Two-Dimensional Figures R ? Practice and Apply For Exercises See Examples 10–13 16, 17, 27–29 18–26 1 3 For each pair of congruent triangles, name the corresponding parts. Then complete the congruence statement. 10. 11. K D P M N 2 Extra Practice See page 748. E F DFE Q J R P KJM ? 12. A 13. D ? Z R B E C DBA T W S Y ? ZWY ? Tell whether each statement is sometimes, always, or never true. 14. If two triangles are congruent, then the perimeters are equal. 15. If the perimeters of two triangles are equal, then the triangles are congruent. ARCHITECTURE For Exercises 16 and 17, use the diagram of the roof truss at the right and the fact that TRU SRU. 16. Find the distance from the left metal plate connector to the center web. R top left chord center web metal plate connector T 30˚ S U bottom chord 32 ft 17. What is the measure of the angle formed by the top left chord and the bottom chord? Architecture Trusses were used in the construction of the Eiffel Tower in Paris, France. The tower contains more than 15,000 pieces of steel and 2.5 million rivets. Source: www.paris.org Complete each congruence statement if FHG CBD and KMA PRQ. 18. G ? 19. C ? 20. M ? 21. Q ? 22. HG ? 23. BC C 24. D ? 25. GF ? 26. GFH ? Find the value of x for each pair of congruent triangles. 27. A 28. D 12 R C 16 ? S J x x B www.pre-alg.com/self_check_quiz 18 20 6 E T K 12 M Lesson 10-2 Congruent Triangles 503 29. ALGEBRA If ABC XYZ, what is the value of x? A 3x 12 A X 24 Y 18 X 30 B Z B M N Q Y QUILTS For Exercises 30–33, use the quilt pattern shown at the left. Name a triangle that appears to be congruent to each triangle listed. 30. XBQ 31. YQP Z O P W C C 32. DWO D 33. ZQN 34. CRITICAL THINKING In the figure at the right, there are two pairs of congruent triangles. Write a congruence statement for each pair. D C E J B K 35. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. Where are congruent triangles present A I H G in nature? Include the following in your answer: • a definition of congruent triangles, and • two examples of objects in nature that contain congruent triangles. Standardized Test Practice F 36. Which of the following must be true if ACD EHF? A C E B CA H F 37. Find the measure of UY if XYZ UYW. A 10 B 20 C 24 D D CE F C D CAD HEF W 10 X Y 26 26 U 24 Z Maintain Your Skills Mixed Review 38. Angles P and Q are supplementary. Find mP if mQ 129°. (Lesson 10-1) Use a calculator to find each value to the nearest ten-thousandth. (Lesson 9-8) 39. cos 83° 40. sin 39.7° 41. tan 49.2° 1 2 42. Name the multiplicative inverse of 2. 43. ALGEBRA Getting Ready for the Next Lesson (Lesson 5-4) Solve 5a 6 24. (Lesson 3-5) PREREQUISITE SKILL Graph each point on a coordinate system. (To review the coordinate system, see Lesson 2-6.) 44. A(2, 4) 45. J(1, 3) 46. H(0, 5) 47. D(2, 0) 48. W(2, 4) 49. T(1, 3) 504 Chapter 10 Two-Dimensional Figures A Preview of Lesson 10-3 Symmetry Activity 1 Trace the outline of the butterfly shown. Then draw a line down the center of the butterfly. Notice how the two halves match. When this happens, a figure is said to have line symmetry and the line is called a line of symmetry. A figure that has line symmetry has bilateral symmetry . Analyze Determine whether each figure has line symmetry. If it does, trace the figure, and draw all lines of symmetry. If not, write none. 1. 2. J 3. Activity 2 Copy the figure at the right. Then cut out the figure. Next, rotate the figure 90°, 180°, and 270°, about point C. What do you notice about the appearance of the figure in each rotation? C Any figure that can be turned or rotated less than 360° about a fixed point so that the figure looks exactly as it does in its original position has rotational or turn symmetry . Analyze Determine whether each figure has rotational symmetry. Write yes or no. 4. 5. 6. 7. Name three objects that have both line symmetry and rotational symmetry. Algebra Activity Symmetry 505 Transformations on the Coordinate Plane • Draw translations, rotations, and reflections on a coordinate plane. are transformations involved in recreational activities? Vocabulary • • • • • transformation translation reflection line of symmetry rotation The physical motions used in recreational activities such as skateboarding, swinging, or riding a scooter, are related to mathematics. a. Describe the motion involved in making a 180° turn on a skateboard. b. Describe the motion that is used when swinging on a swing. c. What type of motion does a scooter display when moving? TRANSFORMATIONS A movement of a geometric figure is a transformation . Three types of transformations are shown below. • In a translation , you slide a figure from one position to another without turning it. Translations are also called slides. Translation y x O • In a reflection , you flip a figure over a line. The figures are mirror images of each other. Reflections are also called flips. The line is called a line of symmetry . Reflection y line of symmetry x O • In a rotation , you turn the figure around a fixed point. Rotations are also called turns. Rotation y O fixed point 506 Chapter 10 Two-Dimensional Figures x Study Tip Common Misconception In a translation, the order in which a figure is moved does not matter. For example, moving 3 units down and then 2 units across is the same as moving 2 units across and then 3 units down. When translating a figure, every point of the original figure is moved the same distance and in the same direction. Translation 5 units down Translation 4 units left y Translation 6 units right, 3 units up y x O y x O x O The following steps can be used to translate a point in the coordinate plane. Translation Step 1 Describe the translation using an ordered pair. Step 2 Add the coordinates of the ordered pair to the coordinates of the original point. Example 1 Translation in a Coordinate Plane Study Tip The vertices of MNP are M(4, 2), N(0, 2), and P(5, 2). Graph the triangle and the image of MNP after a translation 5 units left and 3 units up. This translation can be written as the ordered pair (5, 3). To find the coordinates of the translated image, add 5 to each x-coordinate and add 3 to each y-coordinate. y vertex 5 left, 3 up translation P' N' M(4, 2) (5, 3) → M(1, 1) N P Notation N(0, 2) (5, 3) → N(5, 5) The notation M is read M prime. It corresponds to point M. P(5, 2) (5, 3) → P(0, 5) M' O The coordinates of the vertices of MNP are M(1, 1), N(5, 5), and P(0, 5). x M When reflecting a figure, every point of the original figure has a corresponding point on the other side of the line of symmetry. Reflection over the x-axis Reflection over the y-axis y O www.pre-alg.com/extra_examples y x O x Lesson 10-3 Transformations on the Coordinate Plane 507 The following rules can be used to reflect a point over the x- or y-axis. Reflection • To reflect a point over the x-axis, use the same x-coordinate and multiply the y-coordinate by –1. • To reflect a point over the y-axis, use the same y-coordinate and multiply the x-coordinate by –1. Example 2 Reflection in a Coordinate Plane The vertices of the figure below are A(2, 3), B(0, 5), C(3, 1), and D(3, 3). Graph the figure and the image of the figure after a reflection over the x-axis. To find the coordinates of the vertices of the image after a reflection over the x-axis, use the same x-coordinate and multiply the y-coordinate by –1. vertex reflection A(2, 3) → (2, 1 3) → A(2, 3) B(0, 5) → (0, 1 5) → B(0, 5) C(3, 1) → (3, 1 1) → C(3, 1) D(3, 3) → (3, 1 3) → D(3, 3) y C' x O A' D' B' The coordinates of the vertices of the reflected figure are A(2, 3), B(0, 5), C(3, 1), and D(3, 3). The diagrams below show three of the ways a figure can be rotated. Reading Math Rotation of 90˚ clockwise Rotation of 90˚ counterclockwise y Rotation of 180˚ y y clockwise The word clockwise refers to the direction in which the hands of a clock rotate. O counterclockwise x O x x O The word counterclockwise refers to the direction opposite to that in which the hands of a clock rotate. You can use these rules to rotate a figure 90° clockwise, 90° counterclockwise, or 180° about the origin. Rotation • To rotate a figure 90° clockwise about the origin, switch the coordinates of each point and then multiply the new second coordinate by 1. • To rotate a figure 90° counterclockwise about the origin, switch the coordinates of each point and then multiply the new first coordinate by 1. • To rotate a figure 180° about the origin, multiply both coordinates of each point by 1. 508 Chapter 10 Two-Dimensional Figures Example 3 Rotations in a Coordinate Plane A figure has vertices J(1, 1), K(4, 1), M(1, 2), N(4, 2), and P(3, 4). Graph the figure and the image of the figure after a rotation of 90° counterclockwise. To rotate the figure, switch the coordinates N' K' y P of each vertex and multiply the first by 1. P' J(1, 1) → J(1, 1) N(4, 2) → N(2, 4) K(4, 1) → K(1, 4) P(3, 4) → P(4, 3) M N J M' J' K x O M(1, 2) → M(2, 1) The coordinates of the vertices of the rotated figure are J(1, 1), K(1, 4), M(2, 1), N(2, 4), and P(4, 3). Concept Check 1. Write a sentence to describe a figure that is translated by (5, 2). 2. OPEN ENDED Draw a triangle on grid paper. Then draw the image of the triangle after it is moved 5 units right and then rotated counterclockwise 90°. Guided Practice GUIDED PRACTICE KEY 3. Rectangle RSTU is shown at the right. Graph the image of the rectangle after a translation 4 units right and 2 units down. 4. Suppose the figure graphed is reflected over the y-axis. Find the coordinates of the vertices after the reflection. y R S U T O X x y W x O Y Z 5. Triangle ABC is shown. Graph the image of ABC after a rotation of 90° counterclockwise. y B A O x C Application 6. ART Identify the type of transformation that is shown below. Lesson 10-3 Transformations on the Coordinate Plane 509 Practice and Apply For Exercises See Examples 7–10 11–14 15–17 18–20 1 2 3 1–3 Find the coordinates of the vertices of each figure after the given translation. Then graph the translation image. 1 7. (2, 3) 8. (4, 3) 9. 5, 2 2 y y y E A Extra Practice D B See page 748. H x O x O x O J I K C F G 10. The vertices of a figure are D(1, 2), E(1, 4), F(1, 2), G(4, 4). Graph the image of the figure after a translation 4 units down. Find the coordinates of the vertices of each figure after a reflection over the given axis. Then graph the reflection image. 11. x-axis 12. y-axis 13. x-axis y y R S y B N A M O C x O x O P x OQ D T 14. The vertices of a figure are W(3, 3), X(0, 4), Y(4, 2), and Z(2, 1). Graph the image of its reflection over the y-axis. For Exercises 15–17, use the graph shown. 15. Graph the image of the figure after a rotation of 90° counterclockwise. y N B 16. Find the coordinates of the vertices of the figure after a 180° rotation. x O A C 17. Graph the image of the figure after a rotation of 90° clockwise. E D Identify each transformation as a translation, a reflection, or a rotation. y y y 18. 19. 20. O x O x O x 21. Give a counterexample for the following statement. The image of a figure’s reflection is never the same as the image of its translation. 510 Chapter 10 Two-Dimensional Figures 22. GAMES What type of transformation is used when moving a knight in a game of chess? 23. MIRRORS Which transformation exists when you look into a mirror? Games In chess, each player has 16 game pieces, or chessmen. There are two rooks, two knights, two bishops, a queen, a king, and eight pawns. Source: www.infoplease.com Standardized Test Practice 24. CRITICAL THINKING After a rotation of 90° counterclockwise, the coordinates of the vertices of the image of ABC are A(3, 2), B(0, 4), and C(5, 5). What were the coordinates of the vertices before the rotation? Answer the question that was posed at the beginning of the lesson. How are the transformations involved in recreational activities? Include the following in your answer: • an explanation describing each type of transformation, and • an explanation telling the type of transformation each recreational activity represents. 25. WRITING IN MATH For Exercises 26 and 27, suppose the figure shown is translated 4 units to the left and 3 units down. 26. Which point is not a vertex of the translated image? A (2, 3) B (4, 3) C (3, 1) D y O x (2, 1) 27. Which statement best describes this translation? A (x, y) → (x 4, y 3) B (x, y) → (x 4, y 3) C (x, y) → (x 4, y 3) D (x, y) → (x 4, y 3) Maintain Your Skills Mixed Review Complete each congruence statement if ABC DEF. (Lesson 10-2) 28. D ? 29. AC 30. DE ? ? Find the value of x in each figure. 31. (Lesson 10-1) 32. 110˚ 26˚ x x 33. ALGEBRA Solve x 3.4 6.2. Graph the solution on a number line. (Lesson 7-4) 34. Evaluate 4 3. (Lesson 2-1) Getting Ready for the Next Lesson PREREQUISITE SKILL Solve each equation. (To review solving two-step equations, see Lesson 3-5.) 35. 2x 134 360 36. 3x 54 360 37. 5x 125 360 38. 4x 92 360 39. 2x 148 360 40. 6x 102 360 www.pre-alg.com/self_check_quiz Lesson 10-3 Transformations on the Coordinate Plane 511 A Follow-Up of Lesson 10-3 Dilations In this activity, you will investigate dilations , which alter the size of a figure. Collect the Data Step 1 Draw and label a polygon on a coordinate plane. Trapezoid ABCD is shown. Step 2 Suppose the scale factor is 2. Multiply the coordinates of each vertex by 2. A(1, 2) → A’(2, 4) B(0, 3) → B’(0, 6) C(4, 1) → C’(8, 2) D(2, 1) → D’(4, 2) y B' A' B C' A C x O Step 3 Draw the new trapezoid. D D' Analyze the Data 1. 2. 3. 4. Use a protractor to measure the angles in each trapezoid. How do they compare? Use a ruler to measure the sides of each trapezoid. How do they compare? What ratio compares the measures of the corresponding sides? 1 Repeat the activity by multiplying the coordinates of trapezoid ABCD by . 2 Are the results the same? Explain. Make a Conjecture 5. Explain how you know whether a dilation is a reduction or an enlargement. 6. Explain the difference between dilations and the other types of transformations. Extend the Activity Find the coordinates of the dilation image for the given scale factor, and graph the dilation image. 1 8. 7. 3 4 C y y 1 2 9. 1 y x G O A D M F B O x O x N E Identify each transformation as a translation, rotation, reflection, or dilation. 10. y A y 11. A' Y C C' X J O Y' x O 12. G X' x G' H y K H' O x Z B E' Z' J' K' 512 Investigating 512 Chapter 10 Two-Dimensional Figures Quadrilaterals • Find the missing angle measures of a quadrilateral. • Classify quadrilaterals. Vocabulary • quadrilateral are quadrilaterals used in design? Geometric figures are often used to create various designs. The design of a brick walkway is shown at the right. Notice how the walkway is formed using different-shaped bricks to create circles. a. Describe the bricks that are used to create the smallest circles. b. Describe how the shape of the bricks change as the circles get larger. QUADRILATERALS Squares, rectangles, and trapezoids are examples of quadrilaterals. A quadrilateral is a closed figure with four sides and four vertices. The segments that make up a quadrilateral intersect only at their endpoints. Quadrilaterals Not Quadrilaterals As with triangles, a quadrilateral can be named by its vertices. The quadrilateral below can be named quadrilateral ABCD. Study Tip Naming Quadrilaterals When you name a quadrilateral, you can begin at any vertex. However, it is important to name vertices in order. The vertices are A, B, C, and D. A B The sides are AB, BC, CD, and DA. C The angles are A, B, C, and D. D The quadrilateral shown above has many names. For example, it can also be named quadrilateral BCDA, quadrilateral DABC, or quadrilateral CBAD. Concept Check How many vertices does a quadrilateral have? Lesson 10-4 Quadrilaterals 513 A quadrilateral can be separated into two triangles. Since the sum of the measures of the angles of a triangle is 180°, the sum of the measures of the angles of a quadrilateral is 2(180°) or 360°. F G E H Angles of a Quadrilateral The sum of the measures of the angles of a quadrilateral is 360°. Example 1 Find Angle Measures ALGEBRA Find the value of x. Then find each missing angle measure. Words Variable Study Tip Check Your Work To check the answer, find the sum of the measures of the angles. Since 88° 62° 140° 70° 360˚, the answer is correct. B 62˚ 2x ˚ The sum of the measures of the angles is 360°. A 88˚ C D Let mA, mB, mC, and mD represent the measures of the angles. Equation mA mB mC mD 360 88 62 2x x 360 3x 150 360 3x 150 150 360 150 3x 210 x 70 Angles of a quadrilateral Substitution Combine like terms. Subtract 150 from each side. Simplify. The value of x is 70. So, mD 70° and mC 2(70) or 140°. CLASSIFY QUADRILATERALS The diagram below shows how quadrilaterals are related. Notice that it goes from the most general quadrilateral to the most specific. Quadrilateral Parallelogram Trapezoid quadrilateral with one pair of opposite sides parallel quadrilateral with both pairs of opposite sides parallel and congruent Rhombus parallelogram with 4 congruent sides Rectangle parallelogram with 4 right angles Square parallelogram with 4 congruent sides and 4 right angles The best description of a quadrilateral is the one that is the most specific. 514 Chapter 10 Two-Dimensional Figures x˚ Example 2 Classify Quadrilaterals Classify each quadrilateral using the name that best describes it. a. b. The quadrilateral has four congruent sides and four right angles. It is a square. Artist Visual artists create paintings, sculptures, or illustrations to communicate ideas. The quadrilateral has opposite sides parallel and opposite sides congruent. It is a parallelogram. c. ART Classify the quadrilaterals that are outlined in the painting at the right. Each of the quadrilaterals has four right angles, but the four sides are not congruent. The quadrilaterals are rectangles. Online Research For information about a career as a visual artist, visit: www.pre-alg.com/ careers Irene Rice Periera. Untitled. 1951 Concept Check 1. OPEN ENDED Give a real-world example of a quadrilateral, a parallelogram, a rhombus, and a square. 2. Describe the characteristics of a rectangle and draw an example of one. Guided Practice GUIDED PRACTICE KEY ALGEBRA 3. x˚ 68˚ Find the value of x. Then find the missing angle measures. 4. 125˚ 64˚ 100˚ 118˚ 2x ˚ x˚ 7 ⫺10 ⫺10 8 8 7. SPORTS Classify the quadrilaterals that are found on the scoring region of a shuffleboard court. 10 Application 7 Classify each quadrilateral using the name that best describes it. 5. 6. www.pre-alg.com/extra_examples Lesson 10-4 Quadrilaterals 515 Practice and Apply ALGEBRA For Exercises See Examples 8–13 14–22 1 2 8. 65˚ Find the value of x. Then find the missing angle measures. 9. 109˚ 11. 3x ˚ x˚ 110˚ x˚ See page 748. 128˚ 96˚ 115˚ Extra Practice 10. x˚ x˚ 12. x˚ 2x ˚ 120˚ 52˚ 13. 2x ˚ x˚ 90˚ (2x 20)˚ 120˚ 135˚ (x 5)˚ x˚ (x 10)˚ 14. COOKING Name an item found in a kitchen that is rectangular in shape. Explain why the item is a rectangle. 15. GAMES Identify a board game that is played on a board that is shaped like a square. Explain why the board is a square. Classify each quadrilateral using the name that best describes it. 16. 17. 18. 19. 20. 21. 22. ART The abstract painting is an example of how shape and color are used in art. Write a few sentences describing the geometric shapes used by the artist. Tell whether each statement is sometimes, always, or never true. 23. A square is a rhombus. 24. A parallelogram is a rectangle. 25. A rectangle is a square. 26. A parallelogram is a quadrilateral. Elizabeth Murray. Painter’s Progress, 1981 Make a drawing of each quadrilateral. Then classify each quadrilateral using the name that best describes it. 27. In quadrilateral JKLM, mJ 90°, mK 50°, mL 90°, and mM 130°. 28. In quadrilateral CDEF, CD and E F are parallel, and C F and D E are parallel. Angle C is not congruent to D. 516 Chapter 10 Two-Dimensional Figures 29. CRITICAL THINKING An equilateral figure is one in which all sides have the same measure. An equiangular figure is one in which all angles have the same measure. a. Is it possible for a quadrilateral to be equilateral without being equiangular? If so, explain with a drawing. b. Is it possible for a quadrilateral to be equiangular without being equilateral? If so, explain with a drawing. Answer the question that was posed at the beginning of the lesson. How are quadrilaterals used in design? Include the following in your answer: • an example of a real-world design that contains quadrilaterals, and • an explanation of the figures used in the design. 30. WRITING IN MATH Standardized Test Practice 31. Which figure is a rhombus? A B C 32. GRID IN Find the value of x in the figure at the right. D 82˚ 41˚ x˚ Maintain Your Skills Mixed Review 33. A figure has vertices D(1, 2), E(1, 4), F(4, 4), and G(2, 2). Graph the figure and its image after a translation 4 units down. (Lesson 10-3) Complete each congruence statement if AKM NDQ. (Lesson 10-2) 34. K ? 35. Q N ? 36. Find the discount for a $45 shirt that is on sale for 20% off. (Lesson 6-7) Getting Ready for the Next Lesson PREREQUISITE SKILL Find each product. (To review multiplying decimals, see page 715.) 37. (3)(4.8) 38. (5.4)(6) 39. (9.2)(3.1) P ractice Quiz 1 40. (10.5)(5.7) Lessons 10-1 through 10-4 1. Angle A and B are complementary. Find mA if mB 55°. (Lesson 10-1) 2. Suppose ABC DEF. Which angle is congruent to D? (Lesson 10-2) 3. QRS has vertices Q(3, 3), R(5, 6), and S(7, 3). Find the coordinates for the vertices of the triangle after the figure is reflected over the x-axis. (Lesson 10-3) ALGEBRA 4. 25˚ Find the value of x. Then find the missing angle measures. 5. 2x ˚ x˚ 175˚ 95˚ www.pre-alg.com/self_check_quiz (Lesson 10-4) x˚ Lesson 10-4 Quadrilaterals 517 A Preview of Lesson 10-5 Area and Geoboards Activity 1 One square on a geoboard has an area of one square unit. 5 5 5 4 4 4 3 3 3 2 2 2 1 1 2 3 4 1 5 1 The area is about 5 square units. 2 3 4 1 5 The area is about 6 square units. 1 2 3 4 5 The area is 3.5 square units. Model and Analyze Find the area of each figure. Estimate, if necessary. 1. 2. 5 3. 5 5 4 4 4 3 3 3 2 2 2 1 1 2 3 4 5 1 1 2 3 4 1 5 1 2 3 4 5 4. Explain how you found the area of each figure in Exercises 1–3. 5. Make a figure on the geoboard. Ask a classmate to find the area of the figure. Activity 2 The following example shows how to find the area of a right triangle on a geoboard. Step 1 First, make another triangle so that the two triangles form a rectangle. Step 2 Then find the area of the rectangle. Step 3 Next, divide by 2 to find the area of each triangle. The area of the rectangle is 6 square units. So, the area of each triangle is 3 square units. 518 Investigating Slope-Intercept Form 518 Chapter 10 Two-Dimensional Figures 5 5 4 4 3 3 2 2 1 1 2 3 4 5 1 1 2 3 4 5 Model and Analyze Find the area of each triangle. 6. 7. 5 8. 5 5 4 4 4 3 3 3 2 2 2 1 1 2 3 4 1 5 1 2 3 4 1 5 1 2 3 4 5 2 3 9. Make a right triangle on your geoboard. Find its area using this method. 10. Write a few sentences explaining how you found the area of the triangle. Activity 3 Another way to find the area of a figure on a geoboard is to build a rectangle around the figure. Consider the following example. 5 Step 1 Make the triangle shown on a geoboard. 4 Step 2 Build a rectangle around the triangle. 3 Step 3 Subtract to find the area of the original triangle. 2 1 The area of the rectangle is 16 square units. 5 a 4 The area of triangle b is half of 12 or 6 square units. b 1 4 5 The area of triangle a is half of 4 or 2 square units. 3 2 1 1 2 3 4 5 The area of the original triangle is 16 2 6 or 8 square units. Model and Analyze Find the area of each figure by building a rectangle around the figure. 11. 12. 5 13. 5 5 4 4 4 3 3 3 2 2 2 1 1 2 3 4 5 1 1 2 3 4 5 1 1 14. Make a figure on your geoboard. Find the area using this method. 2 3 4 5 Investigating Algebra Activity Area and Geoboards 519 Area: Parallelograms, Triangles, and Trapezoids • Find area of parallelograms. • Find the areas of triangles and trapezoids. Vocabulary • base • altitude is the area of a parallelogram related to the area of a rectangle? The area of a rectangle can be found by multiplying the length and width. The rectangle shown below has an area of 3 6 or 18 square units. Suppose a triangle is cut from one side of the rectangle and moved to the other side. 3 units 6 units a. What figure is formed? b. Compare the area of the rectangle to the area of the parallelogram. c. What parts of a rectangle and parallelogram determine their area? AREAS OF PARALLELOGRAMS The area of a parallelogram can be found by multiplying the measures of the base and the height. The base can be any side of the parallelogram. The height is the length of an altitude,, a line segment altitude perpendicular to the bases with endpoints on the base and the side opposite the base. altitude base Area of a Parallelogram • Words If a parallelogram has a base of b units and a height of h units, then the area A is bh square units. • Model b h • Symbols A bh Example 1 Find Areas of Parallelograms Find the area of each parallelogram. a. The base is 4 feet. The height is 2 feet. 2 ft 4 ft A bh A42 A8 Area of a parallelogram Replace b with 4 and h with 2. Multiply. The area is 8 square feet. 520 Chapter 10 Two-Dimensional Figures 7.5 cm The base is 5.9 centimeters. The height is 7.5 centimeters. Area of a parallelogram A bh A (5.9)(7.5) Replace b with 5.9 and h with 7.5. A 44.25 Multiply. 5.9 cm The area is 44.25 square centimeters. b. AREA OF TRIANGLES AND TRAPEZOIDS A diagonal of a parallelogram separates the parallelogram into two congruent triangles. The area of each triangle is one-half the area of the parallelogram. diagonal The area of parallelogram ABCD is 7 ⭈ 4 or 28 square units. B C 4 A The area of triangle ABD is 21 ⭈ 28 or 14 square units. D 7 Using the formula for the area of a parallelogram, we can find the formula for the area of a triangle. Area of a Triangle • Words If a triangle has a base of b units and a height of h units, then the 1 area A is bh square units. 1 A bh 2 • Symbols 2 • Model h b Concept Check The area of a triangle is one half of the area of what figure with the same height and base? As with parallelograms, any side of a triangle can be used as a base. The height is the length of a corresponding altitude. E A N B NR is extended to S. P Study Tip Altitudes QR is extended to T. F D T An altitude can be outside the triangle. C Q R S base A C A B B C base N Q N R Q R altitude B D C E A F altitude R P Q S N T Concept Check www.pre-alg.com/extra_examples Which side of a triangle can be used as the base? Lesson 10-5 Area: Parallelograms, Triangles, and Trapezoids 521 Example 2 Find Areas of Triangles Find the area of each triangle. a. The base is 5 inches. The height is 6 inches. 6 in. 5 in. 1 2 1 A (5)(6) 2 1 A (30) 2 A bh Study Tip Alternative Method Multiplication is commutative and associative. So you can 1 2 also find of 6 first and Area of a triangle Replace b with 5 and h with 6. Multiply. 5 6 30 A 15 The area of the triangle is 15 square inches. 4.2 m The base is 7 meters. The height is 4.2 meters. b. then multiply by 5. 7m 1 2 1 A (7)(4.2) 2 1 A (29.4) 2 A bh Area of a triangle Replace b with 7 and h with 4.2. Multiply. 7 4.2 29.4 A 14.7 The area of the triangle is 14.7 square meters. A trapezoid has two bases. The height of a trapezoid is the distance between the bases. A trapezoid can be separated into two triangles. F G a h E J b K The triangles are FGH and EFH. h The measure of a base of EFH is b units. H The altitudes of the triangles, FJ and H K , are congruent. Both are h units long. The measure of a base of FGH is a units. area of trapezoid EFGH area of FGH area of EFH 1 ah 2 1 bh 2 1 2 h(a b) Distributive Property Area of a Trapezoid • Words If a trapezoid has bases of a units and b units and a height of 1 h units, then the area A of the trapezoid is h(a b) 2 square units. • Symbols A 12h(a b) a • Model h b 522 Chapter 10 Two-Dimensional Figures Example 3 Find Area of a Trapezoid 6 1 in. Find the area of the trapezoid. 2 The height is 4 inches. 1 2 1 4 4 in. The bases are 6 inches and 3 inches. 1 2 1 1 1 A 4 6 3 2 2 4 1 3 A 4 · 9 2 4 1 4 39 A 2 1 4 39 1 A or 19 2 2 A h(a b) Study Tip Look Back To review multiplying fractions, see Lesson 5-3. Area of a trapezoid 3 1 in. 4 1 1 Replace h with 4, a with 6 and b with 3. 2 4 1 2 1 4 3 4 6 3 9 Divide out the common factors. 1 2 The area of the trapezoid is 19 square inches. Example 4 Use Area to Solve a Problem FLAGS The signal flag shown represents the number five. Find the area of the blue region. To find the area of the blue region, subtract the areas of the triangles from the area of the square. 10 in. Area of the square Area of each triangle A bh A 14 14 1 2 1 A 10 7 2 A 196 A 35 7 in. 14 in. 14 in. A bh 1 · 10 5, 5 7 35 2 The total area of the triangles is 4(35) or 140 square inches. So, the area of the blue region is 196 140 or 56 square inches. Concept Check 1. OPEN ENDED Draw and label a parallelogram that has an area of 24 square inches. 2. Define an altitude of a triangle and draw an example. Guided Practice Find the area of each figure. 3. 4. 4 ft 5. 6m 5.4 cm 2 ft 8m 3 cm Application 6. FLAGS The flag shown at the right is the international signal for the number three. Find the area of the red region. 15 m 7.6 m 5m 18 in. 9 in. 3 in. 7 in. 12 in. Lesson 10-5 Area: Parallelograms, Triangles, and Trapezoids 523 Practice and Apply For Exercises See Examples 7–20 21–25, 29 1–3 4 Find the area of each figure. 7. 8. 9. 7.4 in. 15 cm 3.5 in. 12 cm 2m 5.5 m Extra Practice See page 749. 10. 11. 12. 20 ft 6.7 cm 5.3 cm 10 in. 12 ft 1 9 5 ft 7.2 in. 11 ft 14 ft 9.9 cm 6 in. Find the area of each figure described. 13. triangle: base, 8 in.; height, 7 in. 14. trapezoid: height, 2 cm; bases, 3 cm, 6 cm 15. parallelogram: base, 3.8 yd; height, 6 yd 16. triangle: base, 9 ft; height, 3.2 ft 17. trapezoid: height, 3.5 m; bases, 10 m and 11 m 18. parallelogram: base, 5.6 km; height, 4.5 km GEOGRAPHY For Exercises 19 and 20, use the approximate measurements to estimate the area of each state. 332 mi 270 mi 19. 20. 287 mi A R K A NS A S OR E GON 235 mi Geography The largest U.S. state is Alaska. It has an area of 615,230 square miles. Rhode Island is the smallest state. It has an area of 1231 square miles. Source: The World Almanac 165 mi Find the area of each figure. 12 km 21. 22. 2 km 8 ft 3m 9m 8m 6 ft 4m 8 km 15 km 8 ft 11 ft 6m LAWNCARE For Exercises 24 and 25, use the diagram shown and the following information. Mrs. Malone plans to fertilize her lawn. The fertilizer she will be using indicates that one bag fertilizes 2000 square feet. 24. Find the area of the lawn. 25. How many bags of fertilizer should she buy? 524 Chapter 10 Two-Dimensional Figures 23. 15 ft 125 ft Lawn Flower bed 15 ft Patio 12 ft 100 ft 84 ft 26. Find the base of a parallelogram with a height of 9.2 meters and an area of 36.8 square meters. 27. Suppose a triangle has an area of 20 square inches and a base of 2.5 inches. What is the measure of the height? 28. A trapezoid has an area of 54 square feet. What is the measure of the height if the bases measure 16 feet and 8 feet? 144.34 ft 29. REAL ESTATE The McLaughlins plan to build a house on the lot shown. If an acre is 43,560 square feet, what percent of an acre is the land? 57.62 ft 90.54 ft 147.08 ft 30. CRITICAL THINKING Explain how the formula for the area of a trapezoid can be used to find the formulas for the areas of parallelograms and triangles. Answer the question that was posed at the beginning of the lesson. How is the area of a parallelogram related to the area of a rectangle? Include the following in your answer: • an explanation telling the similarities and differences between a rectangle and a parallelogram, and • a diagram that shows how the area of a parallelogram is related to the area of a rectangle. 31. WRITING IN MATH Standardized Test Practice 32. Which figure does not have an area of 48 square meters? A 8m B C 8m 3m D 5m 1.6 m 6m 12.8 m 12 m 6.4 m 33. Square X has an area of 9 square feet. The sides of square Y are twice as long as the sides of square X. Find the area of square Y. A 18 ft2 B 36 ft2 C 9 ft2 D 6 ft2 Maintain Your Skills Mixed Review Find the value of x. Then find the missing angle measures. 34. 35. x˚ 60˚ 130˚ 110˚ 60˚ 120˚ 4x ˚ x˚ (Lesson 10-4) 36. Triangle MNP has vertices M(1, 1), N(5, 4), and P(4, 1). Graph the image of MNP after a translation 3 units left and 4 units down. (Lesson 10-3) Find the percent of each number mentally. (Lesson 6-6) 37. 25% of 120 Getting Ready for the Next Lesson 38. 75% of 160 39. 40% of 65 PREREQUISITE SKILL Simplify each expression. (To review order of operations, see Lesson 1-2). 40. (5 2)180 www.pre-alg.com/self_check_quiz 41. (7 2)180 42. (10 2)180 43. (9 2)180 Lesson 10-5 Area: Parallelograms, Triangles, and Trapezoids 525 Learning Mathematics Prefixes The table shows some of the prefixes that are used in mathematics. These prefixes are also used in everyday language. In order to use each prefix correctly, you need to understand its meaning. Prefix quad- pent- hex- hept- oct- dec- Meaning Everyday Words Meaning four quadrennial quadruple quadruplet quadriceps happening every four years a sum four times as great as another one of four offspring born at one birth a muscle with four points of origin five Pentagon pentagram pentathlon pentad headquarters of the Department of Defense a five-pointed star a five-event athletic contest a group of five six hexapod hexagonal hexastich hexangular having six having six a poem of having six seven heptad heptagonal heptarchy heptastich a group of seven having seven sides a government by seven rulers a poem of seven lines eight octopus octet octan octennial ten decade decameter decathlon decare feet sides six lines angles a type of mollusk having eight arms a musical composition for eight instruments occurring every eight days lasting eight years a period of ten years ten meters a ten-event athletic contest a metric unit of area equal to 10 acres Reading to Learn 1. Refer to the table above. For each prefix listed, choose one of the everyday words listed and write a sentence that contains the word. 2. RESEARCH Use the Internet, a dictionary, or another reference source to find a mathematical term that contains each of the prefixes listed. Write the definition of each term. 3. RESEARCH Use the Internet, a dictionary, or another reference source to find a different word that contains each prefix. Then define the term. 526 Chapter 10 Two-Dimensional Figures 526 Investigating Polygons • Classify polygons. Vocabulary • • • • polygon diagonal interior angles regular polygon • Determine the sum of the measures of the interior and exterior angles of a polygon. are polygons used in tessellations? The tiled patterns below are called regular tessellations. Notice how the figures repeat to form patterns that contain no gaps or overlaps. a. Which figure is used to create each tessellation? b. Refer to the diagram at the right. What is vertex the sum of the measures of the angles that surround the vertex? c. Does the sum in part b hold true for the square tessellation? Explain. d. Make a conjecture about the sum of the measures of the angles that surround a vertex in the hexagon tessellation. CLASSIFY POLYGONS A polygon is a simple, closed figure formed by three or more line segments. The figures below are examples of polygons. The line segments meet only at their endpoints. The points of intersection are called vertices. The line segments are called sides. The following figures are not polygons. This is not a polygon because it has a curved side. Concept Check This is not a polygon because it is an open figure. This is not a polygon because the sides overlap. Sketch a different figure that is not a polygon. Lesson 10-6 Polygons 527 Study Tip n-gon Polygons can be classified by the number of sides they have. Number of Sides Name of Polygon Number of Sides Name of Polygon 3 4 5 6 triangle quadrilateral pentagon hexagon 7 8 9 10 heptagon octagon nonagon decagon A polygon with n sides is called an n-gon. For example, an octagon can also be called an 8-gon. Example 1 Classify Polygons Classify each polygon. a. b. The polygon has 8 sides. It is an octagon. The polygon has 6 sides. It is a hexagon. MEASURES OF THE ANGLES OF A POLYGON A diagonal is a line segment in a polygon that joins two nonconsecutive vertices. In the diagram below, all possible diagonals from one vertex are shown. quadrilateral pentagon hexagon heptagon octagon Quadrilateral Pentagon Hexagon Heptagon Octagon Sides 4 5 6 7 8 Diagonals 1 2 3 4 5 Triangles 2 3 4 5 6 Notice that the number of triangles is 2 less than the number of sides. You can use the property of the sum of the measures of the angles of a triangle to find the sum of the measures of the interior angles of any polygon. An interior angle is an angle inside a polygon. Interior Angles of a Polygon If a polygon has n sides, then n 2 triangles are formed. The sum of the degree measures of the interior angles of the polygon is (n 2)180. Example 2 Measures of Interior Angles Find the sum of the measures of the interior angles of a heptagon. A heptagon has 7 sides. Therefore, n 7. (n 2)180 (7 2)180 5(180) or 900 Replace n with 7. Simplify. The sum of the measures of the interior angles of a heptagon is 900°. 528 Chapter 10 Two-Dimensional Figures A regular polygon is a polygon that is equilateral (all sides are congruent) and equiangular (all angles are congruent). Since the angles of a regular polygon are congruent, their measures are equal. Example 3 Find Angle Measure of a Regular Polygon SNOW Snowflakes are some of the most beautiful objects in nature. Notice how they are regular and hexagonal in shape. What is the measure of one interior angle in a snowflake? Step 1 Find the sum of the measures of the angles. Snow Snowflakes are also called snow crystals. It is said that no two snowflakes are alike. They differ from each other in size, lacy structure, and surface markings. Source: www.infoplease.com A hexagon has 6 sides. Therefore, n 6. (n 2)180 (6 2)180 Replace n with 6. 4(180) or 720 Simplify. The sum of the measures of the interior angles is 720°. Step 2 Divide the sum by 6 to find the measure of one angle. 720 6 120 So, the measure of one interior angle in a snowflake is 120°. Concept Check 1. OPEN ENDED Draw a polygon that is both equiangular and equilateral. 2. Tell why a square is a regular polygon. 3. Explain the relationship between the number of sides in a polygon and the number of triangles formed by each of the diagonals. Guided Practice Classify each polygon. Then determine whether it appears to be regular or not regular. 4. 5. 6. Find the sum of the measures of the interior angles of a nonagon. 7. What is the measure of each interior angle of a regular heptagon? Round to the nearest tenth. Application 8. TESSELLATIONS Identify the polygons that are used to create the tessellation shown at the right. www.pre-alg.com/extra_examples Lesson 10-6 Polygons 529 Practice and Apply For Exercises See Examples 9–14, 21 29, 30 15–20 23–28 1 Classify each polygon. Then determine whether it appears to be regular or not regular. 9. 10. 11. 2 3 Extra Practice See page 749. 12. 13. 14. Find the sum of the measures of the interior angles of each polygon. 15. pentagon 16. octagon 17. decagon 18. hexagon 19. 18-gon 20. 23-gon ART For Exercises 21 and 22, use the painting shown. 21. List five polygons used in the painting. 22. RESEARCH The title of the painting mentions the music symbol, clef. Use the Internet or another source to find a drawing of a clef. Is a clef a polygon? Explain. Roy Lichtenstein. Modern Painting with Clef. 1967 Find the measure of an interior angle of each polygon. 23. regular nonagon 24. regular pentagon 25. regular octagon 26. regular decagon 27. regular 12-gon 28. regular 25-gon TESSELLATIONS For Exercises 29 and 30, identify the polygons used to create each tessellation. 29. 30. 31. ART Refer to Exercise 8 on page 529. The tessellation design contains regular polygons. Find the perimeter of the design if the measure of the sides of the 12-gon is 5 centimeters. 32. What is the perimeter of a regular pentagon with sides 4.2 feet long? 1 2 33. Find the perimeter of a regular nonagon having sides 6 inches long. 530 Chapter 10 Two-Dimensional Figures 34. CRITICAL THINKING Copy the dot pattern shown at the right. Then without lifting your pencil from the paper, draw four line segments that connect all of the points. Answer the question that was posed at the beginning of the lesson. How are polygons used in tessellations? Include the following in your answer: • an example of a tessellation in which the pattern is formed using only one type of polygon, and • an example of a tessellation in which the pattern is formed using more than one polygon. 35. WRITING IN MATH Standardized Test Practice 36. GRID IN The measure of one angle of a regular polygon with n sides is 180(n 2) . What is the measure of an interior angle in a regular triangle? n 37. Which figure best represents a regular polygon? A Extending the Lesson B C D EXTERIOR ANGLES When a side of a polygon is extended, an exterior angle is formed. 72˚ exterior angle 72˚ In any polygon, the sum of the measures of the exterior angles, one at each vertex, is 360. 72˚ 72˚ 72˚ Find the measure of each exterior angle of each regular polygon. 38. regular octagon 39. regular triangle 40. regular nonagon 41. regular hexagon 42. regular decagon 43. regular 12-gon Maintain Your Skills Mixed Review Find the area of each figure described. 44. triangle: base, 9 in.; height, 6 in. (Lesson 10-5) 45. trapezoid: height, 3 cm; bases, 4 cm, 8 cm Classify each quadrilateral using the name that best describes it. (Lesson 10-4) 46. 49. ALGEBRA Getting Ready for the Next Lesson 47. 48. Simplify 4.6x 2.5x 9.3x. (Lesson 3-2) PREREQUISITE SKILL Use a calculator to find each product. Round to the nearest tenth. (To review rounding decimals, see page 711.) 50. 4.3 51. 2 5.4 52. 42 53. (2.4)2 www.pre-alg.com/self_check_quiz Lesson 10-6 Polygons 531 A Follow-Up of Lesson 10-6 Tessellations A tessellation is a pattern of repeating figures that fit together with no overlapping or empty spaces. Tessellations can be formed using transformations. Activity 1 Create a tessellation using a translation. Draw a square. Then draw a triangle inside the top of the square as shown. Repeat this pattern unit to create a tessellation. Translate or slide the triangle from the top to the bottom of the square. Step 1 Activity 2 It is sometimes helpful to complete one pattern, cut it out, and trace it for the other pattern units. Step 2 Create a tessellation using a rotation. Draw an equilateral triangle. Then draw another triangle inside the left side of the triangle as shown below. Repeat this pattern unit to create a tessellation. Rotate the triangle so you can trace the change on the side as indicated. Step 1 Step 2 Model Use a translation to create a tessellation for each pattern unit shown. 1. 2. 3. Use a rotation to create a tessellation for each pattern unit shown. 4. 5. 6. 7. Make a tessellation that involves a translation, a rotation, or a combination of the two. 532 Investigating Slope-Intercept Form 532 Chapter 10 Two-Dimensional Figures Circumference and Area: Circles • Find circumference of circles. • Find area of circles. Vocabulary • • • • • • circle diameter center circumference radius (pi) are circumference and diameter related? Coins, paper plates, cookies, and CDs are all examples of objects that are circular in shape. Object a. Collect three different-sized circular objects. Then copy the table shown. 1 2 b. Using a tape measure, measure each 3 distance below to the nearest millimeter. Record your results. • the distance across the circular object through its center (d) • the distance around each circular object (C) d C C d C d c. For each object, find the ratio . Record the results in the table. CIRCUMFERENCE OF CIRCLES A circle is the set of all points in a plane that are the same distance from a given point. Study Tip Pi two generally accepted approximations for . The given point is called the center. The distance around the circle is called the circumference. The distance from the center to any point on the circle is its radius. The relationship you discovered in the activity above is true for all circles. The ratio of the circumference of a circle to its diameter is always equal to 3.1415926… The Greek letter (pi) stands for this number. Using this ratio, you can derive a formula for the circumference of a circle. C d Although is an irrational 22 number, 3.14 and are 7 The distance across the circle through its center is its diameter. C d d d C d The ratio of the circumference to the diameter equals pi. Multiply each side by d. Simplify. Circumference of a Circle • Words The circumference of a circle is equal to its diameter times , or 2 times its radius times . • Model C d r • Symbols C d or C 2r Lesson 10-7 Circumference and Area: Circles 533 Concept Check Which term describes the distance from the center of a circle to any point on the circle? Example 1 Find the Circumference of a Circle Find the circumference of each circle to the nearest tenth. a. Circumference of a circle C d 5 cm C 5 Replace d with 5. C 5 Simplify. This is the exact circumference. Study Tip To estimate the circumference, use a calculator. Calculating with 5 Unless otherwise specified, use a calculator to evaluate expressions involving and then follow any instructions regarding rounding. The circumference is about 15.7 centimeters. b. 2nd 3.2 ft [] ENTER 15.70796327 C 2r C 2 3.2 C 20.1 Circumference of a circle Replace r with 3.2. Simplify. Use a calculator. The circumference is about 20.1 feet. Many situations involve circumference and diameter of circles. Example 2 Use Circumference to Solve a Problem TREES A tree in Madison’s yard was damaged in a storm. She wants to replace the tree with another whose trunk is the same size as the original tree. Suppose the circumference of the original tree was 14 inches. What should be the diameter of the replacement tree? Explore You know the circumference of the original tree. You need to know the diameter of the new tree. Plan Use the formula for the circumference of a circle to find the diameter. Solve C d Circumference of a circle 14 d Replace C with 14. 14 d Divide each side by . 4.5 d Simplify. Use a calculator. The diameter of the tree should be about 4.5 inches. Examine Check the reasonableness of the solution by replacing d with 4.5 in C d. C d C 4.5 C 14.1 Circumference of a circle Replace d with 4.5. Simplify. Use a calculator. The solution is reasonable. 534 Chapter 10 Two-Dimensional Figures AREAS OF CIRCLES A circle can be separated into parts as shown below. The parts can then be arranged to form a figure that resembles a parallelogram. Circumference Radius Circumference Since the circle has an area that is relatively close to the area of the figure, you can use the formula for the area of a parallelogram to find the area of a circle. Study Tip A bh Area of a parallelogram A C r 1 2 1 A 2r r 2 The base of the parallelogram is one-half the circumference, and the radius is the height. Arr A r2 Simplify. Look Back To review exponents, see Lesson 4-2. Replace C with 2r. Replace r r with r2. Area of a Circle • Words The area of a circle is equal to times the square of its radius. • Model r • Symbols A r2 Study Tip Estimation To estimate the area of a circle, square the radius and then multiply by 3. Example 3 Find Areas of Circles Find the area of each circle. Round to the nearest tenth. a. A r2 Area of a circle 2 A6 Replace r with 6. 6 in. A 36 Evaluate 62. A 113.1 Use a calculator. The area is about 113.1 square inches. b. 31 m A r2 A (15.5)2 A 240.25 A 754.8 Area of a circle Replace r with 15.5. Evaluate (15.5)2. Use a calculator. The area is about 754.8 square meters. Concept Check 1. Tell how to find the circumference of a circle if you know the measure of the radius. 2. OPEN ENDED Draw and label a circle that has an area between 5 and 8 square units. www.pre-alg.com/extra_examples Lesson 10-7 Circumference and Area: Circles 535 3. FIND THE ERROR Dario and Mark are finding the area of a circle with diameter 7. Dario A 153.9 units2 Mark A 38.5 units 2 Who is correct? Explain your reasoning. Guided Practice Find the circumference and area of each circle. Round to the nearest tenth. 4. 5. 6. 5 mi 4 in. 8m 7. The radius is 1.3 kilometers. Application 8. The diameter is 6.1 centimeters. 9. MUSIC During a football game, a marching band can be heard within a radius of 1.7 miles. What is the area of the neighborhood that can hear the band? Practice and Apply For Exercises See Examples 10–19, 24, 26–35 20–23, 25 1, 3 Find the circumference and area of each circle. Round to the nearest tenth. 10. 11. 12. 13 in. 6 cm 10 m 2 Extra Practice See page 749. 13. 14. 21 km 15. 1 9 2 ft 12.7 m 16. The radius is 4.5 meters. 17. The diameter is 7.3 centimeters. 4 18. The diameter is 7 feet. 5 19. The radius is 15 inches. 3 8 20. What is the diameter of a circle if its circumference is 25.8 inches? Round to the nearest tenth. 21. Find the radius of a circle if its circumference is 9.2 meters. Round to the nearest tenth. 22. Find the radius of a circle if its area is 254.5 square inches. 23. What is the diameter of a circle if its area is 132.7 square meters? 24. BICYCLES If a bicycle tire has a diameter of 27 inches, what is the distance the bicycle will travel in 10 rotations of the tire? 25. SCIENCE The circumference of Earth is about 25,000 miles. What is the distance to the center of Earth? 536 Chapter 10 Two-Dimensional Figures 25,000 mi Match each circle described in the column on the left with its corresponding measurement in the column on the right. 26. radius: 4 units a. circumference: 37.7 units History The Houston Astrodome opened on April 12, 1965. It is the first enclosed stadium built for baseball. The Houston Astrodome has a diameter of 216.4 meters. Source: Compton’s Encyclopedia 27. diameter: 7 units b. area: 7.1 units2 28. diameter: 3 units c. area: 50.3 units2 29. radius: 6 units d. circumference: 22.0 units 30. MONUMENTS The Stonehenge monument in England is enclosed within a circular ditch that has a diameter of 300 feet. Find the area within the ditch to the nearest tenth. 31. HISTORY The dome of the Roman Pantheon has a diameter of 42.7 meters. Use the information at the left to find about how many times more area the Astrodome covers than the Pantheon. FOOD For Exercises 32 and 33, use the following information and the graphic shown at the right. Suppose the circle graph is redrawn onto a poster board so that the diameter of the graph is 9 inches. 32. How much space on the poster board will the circle graph cover? USA TODAY Snapshots® Engineered food Adults who say they believe genetically modified foods are safe as part of our food supply and to the environment: Are safe 40% Don’t know 22% Are unsafe 38% 33. How much of the total space will each section of the graph cover? Source: Les Dames d’Escoffier New York By Cindy Hall and Quin Tian, USA TODAY 34. FUNCTIONS Graph the circumference of a circle as a function of the diameter. Use values of d like 1, 2, 3, 4, and so on. What is the slope of this graph? (Hint: To review slope, see Lesson 8-4.) 35. CRITICAL THINKING The numerical value of the area of a circle is twice the numerical value of the circumference. What is the radius of the circle? (Hint: Use a chart of values for radius, circumference, and area.) Answer the question that was posed at the beginning of the lesson. How are circumference and diameter related? Include the following in your answer: • the ratio of the circumference to the diameter, and • an explanation describing what happens to the circumference as the diameter increases or decreases. 36. WRITING IN MATH Standardized Test Practice 37. The diameter of a circle is 8 units. What is the area of the circle if the diameter is doubled? A 50.3 units2 B 100.5 units2 C 201.1 units2 D 804.2 units2 38. GRID IN The circumference of a circle is 18.8 meters. What is its area to the nearest tenth? www.pre-alg.com/self_check_quiz Lesson 10-7 Circumference and Area: Circles 537 Extending the Lesson B CENTRAL ANGLES A central angle is an angle whose vertex is major arc DA A the center of the circle. It separates E a circle into a major arc and a inscribed angle ACB minor arc. An inscribed angle C minor arc AD has its vertex on the circle and sides that are chords. A chord is central angle DEC D a segment of a circle whose endpoints are on the circle. • The degree measure of a minor arc is the degree measure of the central angle. • The measure of an inscribed angle equals one-half the measure of its intercepted arc. • The degree measure of a major arc is 360 minus the degree measure of the central angle. Refer to the diagram shown. Find the measures of the following angles and arcs. 39. minor arc PQ 40. 1 41. major arc QP 42. 2 43. minor TR 44. minor RQ 45. PSQ 46. minor SR T S P 2 1 125˚ 15˚ R Q 47. List three chords of the circle. Maintain Your Skills Mixed Review Find the measure of an interior angle of each polygon. (Lesson 10-6) 48. regular hexagon 49. regular decagon 50. regular octagon Find the area of each figure described. (Lesson 10-5) 51. trapezoid: height, 2 m; bases, 20 m and 18 m 52. parallelogram: base, 6 km; height, 8 km 53. ALGEBRA Getting Ready for the Next Lesson Solve 2x 7 5x 14. (Lesson 7-6) PREREQUISITE SKILL Find each sum. (To review adding decimals, see page 713.) 54. 200 43.9 55. 23.6 126.9 56. 345.14 23.8 P ractice Quiz 2 Find the area of each figure. 1. Lessons 10-5 through 10-7 (Lesson 10-5) 2. 3. 6m 13.2 cm 7 in. 4m 11 in. 17.1 cm 4. Find the sum of the measures of the interior angles of a 15-gon. (Lesson 10-6) 5. A circle has a radius of 4.7 inches. Find the circumference and area to the nearest tenth. (Lesson 10-7) 538 Chapter 10 Two-Dimensional Figures 10 m Area: Irregular Figures • Find area of irregular figures. can polygons help to find the area of an irregular figure? California is the most populous state in the United States. It ranks third among the U.S. states in area. 210 mi 213.3 mi Source: www.infoplease.com 546.7 mi In the diagram, the area of California is separated into polygons. a. Identify the polygons. b. Explain how polygons can be used to estimate the total land area. c. What is the area of each region? d. What is the total area? 280 mi 133.3 mi 160 mi 40 mi 160 mi AREA OF IRREGULAR FIGURES So far in this chapter, we have discussed the following area formulas. Triangle Trapezoid Parallelogram Circle 1 A bh 2 1 A h(a b) 2 A bh A r2 These formulas can be used to help you find the area of irregular figures. Some examples of irregular figures are shown. I DA HO Study Tip Irregular Figures There can be more than one way to separate an irregular figure. For example, another way to separate the first figure at the right is shown below. To find the area of an irregular figure, separate the irregular figure into figures whose areas you know how to find. parallelogram half of a circle or semicircle triangle I DA HO trapezoid rectangle rectangle Lesson 10-8 Area: Irregular Figures 539 Concept Check Name three area formulas that can be used to find the area of an irregular figure. Example 1 Find Area of Irregular Figures Find the area of the figure to the nearest tenth. 10 cm 12 cm 25 cm Explore You know the dimensions of the figure. You need to find its area. Plan Solve a simpler problem. First, separate the figure into a parallelogram and a semicircle. Then find the sum of the areas of the figures. Estimate: The area of the entire figure should be a little greater than the area of the rectangle. One estimate is 10 25 or 250. Solve Area of Parallelogram A bh Area of a parallelogram A 25 12 Replace b with 25 and h with 12. A 300 Simplify. Study Tip Area of Semicircle Look Back A r2 1 2 1 A 52 2 To review diameter and radius, see Lesson 10-7. A 39.3 Area of a semicircle Replace r with 5. Simplify. The area of the figure is 300 39.3 or about 339.3 square centimeters. Examine Check the reasonableness of the solution by solving the problem another way. Separate the figure into two rectangles and a semicircle. 15 cm 10 cm 12 cm 25 cm The area of one rectangle is 10 12 or 120 square centimeters, the area of the other rectangle is 12 15 or 180 square centimeters, and the area of the semicircle remains 39.3 square centimeters. 120 180 39.3 339.3 So, the answer is correct. 540 Chapter 10 Two-Dimensional Figures Many real-world situations involve finding the area of an irregular figure. Example 2 Use Area of Irregular Figures LANDSCAPE DESIGN Suppose one bag of mulch covers an area of about 9 square feet. How many bags of mulch will be needed to cover the flower garden? 28 ft Step 1 Find the area of the flower garden. Area of rectangle A bh Area of a rectangle A 24 38 Replace b with 24 and h with 38. A 912 Simplify. 38 ft 8 ft Area of parallelogram A bh Area of a parallelogram A 28 8 Replace b with 28 and h with 8. A 224 Simplify. 24 ft The area of the garden is 912 224 or 1136 square feet. Step 2 Find the number of bags of mulch needed. 1136 9 126.2 So, 127 bags of mulch will be needed. Concept Check 1. OPEN ENDED Draw two examples of irregular figures. 2. Write the steps you would use to find the area of the irregular figure shown at the right. Guided Practice Find the area of each figure. Round to the nearest tenth. 3. 4. 5 in. 3 yd 4 yd 5 in. 9 yd Application HOME IMPROVEMENT For Exercises 5 and 6, 9 ft use the diagram shown and the following 7 ft 12 ft information. 20 ft 10 ft The Slavens are planning to stain their wood deck. One gallon of stain costs $19.95 and covers 18 ft approximately 200 square feet. 5. Suppose the Slavens need to apply only one coat of stain. How many gallons of stain will they need to buy? 6. Find the total cost of the stain, not including tax. www.pre-alg.com/extra_examples Lesson 10-8 Area: Irregular Figures 541 Practice and Apply For Exercises See Examples 7–15 19–22 1 2 Find the area of each figure to the nearest tenth, if necessary. 3m 5m 7. 4 ft 8. 9. 4 cm 3m 6 ft 9.5 cm 4 ft Extra Practice 4.2 m See page 750. 12 ft 10. 3.8 cm 11. 5 km 3.2 in. 3.2 in. 4.3 km 10 km 12. 13. 17 m 6 ft 1.5 ft 3m 8 ft 8m 12 m 3 ft 4m 14. What is the area of a figure that is formed using a square with sides 8 meters and a semicircle with a diameter of 5.6 meters? 15. Find the area of a figure formed using a rectangle with base 3.5 yards and height 2.8 yards and a semicircle with radius 7 yards. Find the area of each shaded region. Round to the nearest tenth, if necessary. (Hint: Find the total area and subtract the non-shaded area.) 10 15 16. 17. 18. 7 5 7 7 15 12 8 11 3 4 8 3 14 19. SPORTS In the diagram shown, a track surrounds a football field. To the nearest tenth, what is the area of the grass region inside the track? 20. WALKWAYS A sidewalk forms a 3-foot wide border with the grass as shown. Find the area of the sidewalk. 542 Chapter 10 Two-Dimensional Figures 100 yd 50 50 yd 25 yd 50 12 ft 12 ft The formulas for area and circumference of circles are used in the design of various buildings. Visit www.pre-alg.com/ webquest to continue work on your WebQuest project. GEOGRAPHY For Exercises 21–23, use the diagram shown at the right. 21. Tell how you would separate the irregular figure into polygons to find its area. 170 mi 35 mi 130 mi 290 mi Oklahoma Oklahoma City 22. Use your method to find the total land area of Oklahoma. 225 mi 305 mi 23. RESEARCH Use the Internet or another source to find the actual total land area of Oklahoma. Online Research Data Update How can you use irregular figures to estimate the total land area of your state? Visit www.pre-alg.com/data_update to learn more. 24. CRITICAL THINKING In the diagram, a patio that is 4 feet wide surrounds a swimming pool. What is the area of the patio? 30 ft 24 ft Answer the question that was posed at the beginning of the lesson. How can polygons help to find the area of an irregular figure? Include the following in your answer: • an example of an irregular figure, and • an explanation as to how the figure can be separated to find its area. 25. WRITING IN MATH Standardized Test Practice For Exercises 26 and 27, refer to the diagram shown. Suppose 1 square unit equals 5 square feet. 26. What is the area of the figure? A 26.5 ft2 B 34.5 ft2 C 132.5 ft2 D 185 O ft2 27. What is the area of the nonshaded region? A 73.5 ft2 B 140.5 ft2 C 367.5 ft2 D 473.5 ft2 Maintain Your Skills Mixed Review Find the circumference and area of each circle. Round to the nearest tenth. (Lesson 10-7) 28. The wheel on a game show has a diameter of 8.5 feet. 29. The radius is 7 centimeters. 30. The diameter is 19 inches. Find the sum of the measures of the interior angles of each polygon. (Lesson 10-6) 31. pentagon 32. quadrilateral 33. octagon 34. Draw an angle that measures 35°. (Lesson 9-3) 35. Simplify (4x)(6y). (Lesson 3-2) www.pre-alg.com/self_check_quiz Lesson 10-8 Area: Irregular Figures 543 Vocabulary and Concept Check adjacent angles (p. 493) alternate exterior angles (p. 492) alternate interior angles (p. 492) altitude (p. 520) base (p. 520) bilateral symmetry (p. 505) center (p. 533) circle (p. 533) circumference (p. 533) complementary (p. 493) congruent (p. 500) corresponding angles (p. 492) corresponding parts (p. 500) diagonal (p. 528) diameter (p. 533) dilation (p. 512) exterior angles (p. 492, 531) interior angles (pp. 492, 528) line symmetry (p. 505) line of symmetry (pp. 505, 506) parallel lines (p. 492) parallelogram (p. 514) perpendicular lines (p. 494) pi (p. 533) polygon (p. 527) quadrilateral (p. 513) radius (p. 533) reflection (p. 506) regular polygon (p. 529) rhombus (p. 514) rotation (p. 506) rotational symmetry (p. 505) supplementary (p. 494) tessellation (p. 532) transformation (p. 506) translation (p. 506) transversal (p. 492) trapezoid (p. 514) turn symmetry (p. 505) vertical angles (p. 493) Choose the correct term to complete each sentence. 1. Two angles are (complementary, supplementary ) if the sum of their measures is 180°. 2. A ( rhombus , trapezoid) has four congruent sides. 3. In congruent triangles, the ( corresponding angles , adjacent angles) are congruent. 4. In a ( rotation , translation), a figure is turned around a fixed point. 5. A polygon in which all sides are congruent is called (equiangular, equilateral ). 10-1 Angle Relationships See pages 492–497. Concept Summary • When two parallel lines are cut by a transversal, the corresponding angles, the alternate interior angles, and the alternate exterior angles are congruent. Example • Two angles are complementary if the sum of their measures is 90°. • Two angles are supplementary if the sum of their measures is 180°. t In the figure at the right, m and t is a transversal. If m1 109°, find m7. Since 1 and 7 are alternate exterior angles, they are congruent. So, m7 109°. 1 4 5 8 Exercises Use the figure shown to find the measure of each angle. See Example 1 on page 493. 6. 5 7. 3 8. 2 544 Chapter 10 Two-Dimensional Figures 2 3 6 7 9. 6 m Chapter 10 Study Guide and Review 10-2 Congruent Triangles See pages 500–504. Concept Summary A X • Figures that have the same size and shape are congruent. • The corresponding parts of congruent triangles are congruent. Example B C Y Z ABC XYZ Use ABC and XYZ above to complete each congruence statement. B ? YZ ? B corresponds to Y, so B Y. Y BC BC Z corresponds to , so Y Z . Exercises Complete each congruence statement if FGH QRS. See Example 2 on page 501. 10. F 13. G H 11. S 14. HF ? ? 12. R 15. RQ ? ? ? ? 10-3 Transformations on the Coordinate Plane See pages 506–511. Concept Summary • Three types of transformations are translations, reflections, and rotations. Examples 1 The vertices of JKL are J(1, 2), K(3, 2), and L(1, 1). Graph the triangle and its image after a translation 3 units left and 2 units up. y K' This translation can be written as (3, 2). J' vertex J(1, 2) K(3, 2) L(1, 1) 3 left, 2 up (3, 2) (3, 2) (3, 2) translation → J(2, 4) → K(0, 4) → L(2, 1) J K L' x O L 2 The vertices of figure ABCD are A(1, 3), B(4, 3), C(1, 1), and D(2, 1). Graph the figure and its image after a reflection over the x-axis. Use the same x-coordinate and multiply the y-coordinate by 1. vertex A(1, 3) B(4, 3) C(1, 1) D(2, 1) → → → → (1, 3 1) (4, 3 1) (1, 1 1) (2, 1 1) → → → → reflection A(1, 3) B(4, 3) C(1, 1) D(2, 1) D' D y A' B' C' C O A x B Chapter 10 Study Guide and Review 545 Chapter 10 Study Guide and Review Example 3 A triangle has vertices T(2, 0), W(4, 2), and Z(3, 4). Graph the triangle and its image after a rotation of 180° about the origin. y Multiply both coordinates of each point by 1. Z' vertex T(2, 0) → W(4, 2) → Z(3, 4) → W' rotation T(2, 0) W(4, 2) Z(3, 4) T O x T' W Z Exercises Graph each figure and its image. See Examples 1–3 on pages 507–509. 16. The vertices of a rectangle are C(0, 2), D(2, 0), F(1, 3), and G(3, 1). The rectangle is translated 4 units right and two units down. 17. The vertices of a triangle are H(1, 4), I(4, 2), and J(2, 1). The triangle is reflected over the y-axis. 18. A triangle has vertices N(1, 3), P(3, 0), and Q(1, 1). The triangle is rotated 90° clockwise. 10-4 Quadrilaterals See pages 513–517. Concept Summary • The sum of the angle measures of a quadrilateral is 360°. • A trapezoid, parallelogram, rhombus, square, and rectangle are examples of quadrilaterals. Example Find the value of x. Then find the missing angle measures. x 2x 96 87 360 Angles of a quadrilateral 3x 183 360 Combine like terms. 3x 177 Simplify. x 59 Divide each side by 3. x˚ 2x ˚ 96˚ 87˚ The value of x is 59. So, the missing angle measures are 59° and 2(59) or 118° Exercises Find the value of x. Then find the missing angle measures. See Example 1 on page 514. 19. x˚ 67˚ 66˚ 115˚ 546 Chapter 10 Two-Dimensional Figures 20. 21. 110˚ 2x ˚ 78˚ x˚ x˚ 110˚ x˚ Chapter 10 Study Guide and Review 10-5 Area: Parallelograms, Triangles, and Trapezoids See pages 520–525. Concept Summary • Area of a parallelogram: A bh 1 • Area of a triangle: A 2bh 1 • Area of a trapezoid: A 2h(a b) Example Find the area of the trapezoid. 1 A h(a b) 2 1 A (1.8)(2 5) 2 1 A 1.8 7 2 A 6.3 2 cm Area of a trapezoid 1.8 cm Substitution 5 cm Add 2 and 5. The area of the trapezoid is 6.3 square centimeters. Exercises 22. Find the area of each figure. 23. 13 in. See Examples 1–3 on pages 520–523. 24. 8.7 m 1 5 2 yd 9 in. 6.2 m 4 yd 5.0 m 10-6 Polygons See pages 527–531. Concept Summary • Polygons can be classified by the number of sides they have. • If a polygon has n sides, then the sum of the degree measures of the interior angles of the polygon is (n 2)180. Example Classify the polygon. Then find the sum of the measures of the interior angles. The polygon has 5 sides. It is a pentagon. (n 2)180 (5 2)180 Replace n with 5. 3(180) or 540 Simplify. The sum of the measures of the angles is 540. Exercises Classify each polygon. Then find the sum of the measures of the interior angles. See Examples 1 and 2 on page 528. 25. 26. 27. Chapter 10 Study Guide and Review 547 • Extra Practice, see pages 747–750. • Mixed Problem Solving, see page 767. 10-7 Circumference and Area: Circles See pages 533–538. Example Concept Summary • The circumference C of a circle with radius r is given by C 2r. • The area A of a circle with radius r is given by A r2. Find the circumference and area of the circle. Round to the nearest tenth. C 2r Circumference of a circle C 2 7.5 Replace r with 7.5. C 47.1 The circumference is about 47.1 inches. A r2 A 7.52 A 56.25 A 176.7 15 m Area of a circle Replace r with 7.5. Evaluate 7.52. The area is about 176.7 square meters. Exercises Find the circumference and area of each circle. Round to the nearest tenth. See Examples 1 and 3 on pages 534 and 535. 28. 29. 30. 18 ft 2.1 m 5 cm 10-8 Area: Irregular Figures See pages 539–543. Concept Summary • To find the area of an irregular figure, separate the irregular figure into figures whose areas you know how to find. Example Find the area of the figure. Area of Parallelogram A bh A 3(5) or 15 5 cm Area of Square A s2 A 32 or 9 The area of the figure is 15 9 or 24 square centimeters. 3 cm 3 cm Exercises Find the area of each figure. Round to the nearest tenth, if necessary. See Example 1 on page 540. 11 m 31. 32. 33. 7 in. 34 cm 7 in. 6 in. 16 in. 9.4 m 15 cm 5.2 m 548 Chapter 10 Two-Dimensional Figures Vocabulary and Concepts 1. Draw and label a diagram that represents each of the following. a. congruent triangles b. quadrilateral c. circumference 2. Compare and contrast complementary and supplementary angles. Skills and Applications In the figure at the right, a b, and c is a transversal. If m5 58°, find the measure of each angle. 3. 6 4. 7 5. 4 6. 3 Complete each congruence statement if MNO PRS. 7. P ? 8. R S ? 9. MNO a 2 5 6 8 b 3 1 4 7 c ? Find the coordinates of the vertices of each figure after the given transformation. Then graph the transformation image. 10. reflection over the y-axis 11. rotation of 90° clockwise y E y x O D x O E F G F G H ALGEBRA Find the value of x. Then find the missing angle measure. 12. 13. x˚ ( 2x 1)˚ x ˚ 63˚ 121˚ 70˚ Find the area of each figure described. 14. triangle: base, 21 ft.; height, 16 ft (x 11)˚ 40˚ 15. parallelogram: base, 7 ft; height, 2.5 ft Classify each polygon. Find the sum of the measures of the interior angles. 16. 17. Find the area and circumference of each circle. Round to the nearest tenth. 18. The radius is 3 miles. 19. The diameter is 10 inches. 20. STANDARDIZED TEST PRACTICE What is the diameter of a circle if its circumference is 54.8 meters? Round to the nearest tenth. A 8.7 m B 15.6 m C 17.4 m D 34.9 m www.pre-alg.com/chapter_test Chapter 10 Practice Test 549 Part 1 Multiple Choice Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 55 5 1. Which expression is equivalent to 3 ? (Lessons 4-2 and 4-7) A 52 B 52 C 58 D 515 2. The rectangles shown below are similar. Which proportion can be used to find x? (Lesson 6-2) 6. Triangle ABC is a right triangle. What is the length of the hypotenuse? 12 units B 32 units C 60 units D 84 units 36 B 48 (Lesson 9-5) A A C 7. A graphic artist has designed the logo shown below. If rectangles A and B are parallel, then which statement is true? (Lesson 10-1) A 16 ft 1 x ft 4 ft 2 B 3 ft A C 4 x 16 3 16 x 3 4 B D 3 4 16 x 16 x 4 3 3. Suppose 2% of the containers made by a manufacturer are defective. If 750 of the containers are inspected, how many can be expected not to be defective? (Lesson 6-9) A 150 B 730 C 735 D 748 4. Lawanda plans to paint each side of a cube either white or blue so that when the cube is tossed, the probability that the cube will land 1 3 on a blue side is . How many sides of the A m1 m2 0° B m1 m2 45° C m1 m2 90° D m1 m2 180° 8. In quadrilateral ABCD, mA 100°, mB 100°, and mC 90°. Find mD. (Lesson 10-4) A 70° B 85° C 90° D 170° 9. Refer to the figure shown. What is the best estimate of the area of the circle inscribed in the square? (Lesson 10-7) A 7 in2 B 9 in2 C 19 in2 D 28 in2 cube should she paint blue? (Lesson 6-9) A 1 B 2 C 3 D 4 5. In the spreadsheet below, a formula applied to the values in columns A and B results in the values in column C. What is the formula? 3 in. (Lesson 8-1) A CA–B B C A – 2B C D A B C 1 4 0 4 CAB 2 5 1 3 C A 2B 3 6 2 2 4 7 3 1 550 Chapter 10 Two-Dimensional Figures Test-Taking Tip Question 9 Most standardized tests include any necessary formulas in the test booklet. It helps to be familiar with formulas such as the area of a rectangle and the circumference of a circle, but use any formulas that are given to you. Preparing for Standardized Tests For test-taking strategies and more practice, see pages 771–788. Part 2 Short Response/Grid In Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 14x 6 10. If x 3, what is the value of ? 5x 3 (Lesson 1-2) 11. Refer to the data set shown. 20 20 21 22 24 24 24 26 Which is greater, the median or the mode? (Lesson 5-8) 12. A magazine asked 512 students who use computers if they use e-mail. About 83% of the students said that they use e-mail. About how many of these students use e-mail? (Lesson 6-9) 13. Gracia is finding three consecutive whole numbers whose sum is 78. She uses the equation n (n 1) (n 2) 78. What expression represents the greatest of the three numbers? (Lesson 7-2) 14. Solve 4x 7 39. (Lesson 7-6) 15. What is the slope of the line represented by the equation 3x y 5? (Lesson 8-4) 16. Find the coordinates of the vertices for quadrilateral WXYZ after a translation of (2, 1). (Lesson 10-3) y W X Z Y x O 17. These two triangles are congruent. What is the value of x? (Lesson 9-4) 105˚ 35˚ x˚ www.pre-alg.com/standardized_test 18. What name best classifies this quadrilateral? (Lesson 10-4) 19. The diagram shows a kitchen counter with an area cut out for a sink. What is the area of the counter top? 2 ft 6 ft 3 ft 3 ft 2 ft 12 ft (Lesson 10-8) Part 3 Extended Response Record your answers on a sheet of paper. Show your work. 20. Triangle JKM is shown. (Lesson 10-3) y J M a. What are the K x O coordinates of the vertices of JKM? b. Copy JKM onto a sheet of grid paper. Label the vertices. c. Graph the image of JKM after a translation 2 units left and 3 units down. Label the translated image JKM. d. Graph the image of JKM after a reflection over the y-axis. Label the reflection of JKM as JKM. e. On another sheet of grid paper, graph JKM. Then graph JKM after a reflection over the y-axis. Label the reflection JKM. f. Graph the image of the reflection in part e after a translation 2 units right and 3 units down. Label the translated image JKM. g. Tell whether the image of JKM in part d is the same as the image of JKM in part e. Explain why or why not. Chapter 10 Standardized Test Practice 551