MAC 2.4 Similar Polygons Mac2
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4-7 MAIN IDEA Identify similar polygons and find missing measures of similar polygons. Similar Polygons Follow the steps below to discover how the triangles at the right are related. New Vocabulary polygon similar corresponding parts congruent scale factor Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz ' Copy both triangles onto tracing paper. % Measure and record the sides of each triangle. + & Cut out both triangles. 1. Compare the angles of the , triangles by matching them up. Identify the angle pairs that have equal measure. - 2. Express the ratios _, _, and _ as decimals to the nearest tenth. DF EF LK JK DE LJ 3. What do you notice about the ratios of these sides of matching triangles? A polygon consists of a sequence of consecutive line segments in a plane, placed end to end to form a simple closed figure. Polygons that have the same shape are called similar polygons. In the figure below, polygon ABCD is similar to polygon WXYZ. This is written as polygon ABCD ∼ polygon WXYZ. # 9 " 8 $ : ; % The parts of similar figures that “match” are called corresponding parts. 9 9 8 # " 8 : ; # " $ 218 : ; $ % % Corresponding Angles A W, B X, C Y, D Z Corresponding Sides AB WX, BC XY, CD YZ, DA ZW Chapter 4 Proportions and Similarity The similar triangles in the Mini Lab suggest the following. Similar Polygons Words Key Concept If two polygons are similar, then • their corresponding angles are congruent, or have the same measure, and • the measures of their corresponding sides are proportional. # : Model Reading Math Congruence The symbol is read is congruent to. Arcs are used to show congruent angles. ABC ∼ XYZ " Symbols 9 $ ; ∠A ∠X, ∠B ∠Y, ∠C ∠Z, and BC AC AB _ =_=_ XY YZ XZ Identify Similar Polygons 1 Determine whether rectangle HJKL is similar to rectangle MNPQ. Explain. NP 10 2 , 10 / 6 6 10 2 KL 7 _ =_ 1 3 LH 1 _ =_ or _ 6 2 QM 7 1 Since _ and _ are not equivalent ratios, rectangle HJKL is not similar 2 10 MN 6 7 . Next, check to see if corresponding sides are proportional. JK 3 1 _ =_ or _ 3 - Since the two polygons are rectangles, all of their angles are right angles. Therefore, all corresponding angles are congruent. HJ 7 _ =_ + 3 First, check to see if corresponding angles are congruent. Common Error Do not assume that two polygons are similar just because their corresponding angles are congruent. Their corresponding sides must also be proportional. 7 ) 10 PQ to rectangle MNPQ. Determine whether these polygons are similar. Explain. a. 8 6 b. 12 6 8 8 " 6 14 % # 14 6 + 3.5 1.5 . , 1.5 3.5 - $ The ratio of the lengths of two corresponding sides of two similar polygons is called the scale factor. You can use the scale factor of similar figures or a proportion to find missing measures. Lesson 4-7 Similar Polygons 219 Find Missing Measures 8 " 2 GEOMETRY Given that polygon WXYZ ∼ polygon ABCD, find the missing measure. 24 # 13 12 m % 10 $ ; METHOD 1 9 15 : Write a proportion. −− The missing measure m is the length of XY. Write a proportion. Reading Math Segment Measure The −− measure of XY is written as XY. It represents a number. XY YZ _ =_ polygon WXYZ polygon ABCD BC polygon WXYZ polygon ABCD 15 m _ =_ XY = m, BC = 12, YZ = 15, and CD = 10. m · 10 = 12 · 15 10m = 180 Find the cross products. m = 18 Divide each side by 10. 12 METHOD 2 CD 10 Multiply. Use the scale factor to write an equation. Find the scale factor from polygon WXYZ to polygon ABCD. 15 3 YZ =_ or _ scale factor: _ CD 2 10 A length on Words polygon WXYZ Equation _ _ _3 times as 2 long as m= 3 m=_ (12) Write the equation. m = 18 Multiply. 2 is a corresponding length on polygon ABCD. −− Let m represent the measure of XY. Variable Scale Factor In Example 2, the scale factor from polygon ABCD to polygon WXYZ is 32 , which means that a length on polygon ABCD is 32 as long as a length on polygon WXYZ. The scale factor is the constant of proportionality. _3 · 12 2 Find each missing measure above. c. WZ d. AB Square A ∼ square B with a scale factor of 3:2. Notice that the ratio of their perimeters is 12:8 or 3:2. 3m Square A 220 Chapter 4 Proportions and Similarity 2m Square B Square Perimeter A 12 m B 8m This and other related examples suggest the following. Ratios of Similar Figures Words Key Concept If two figures are similar with Model a a scale factor of _, then the a b perimeters of the figures have a ratio of _a . Figure A Figure B b 3 Triangle LMN is similar Similarity Statements In naming similar triangles, the order of the vertices indicates the corresponding parts. Read the similarity statement carefully to be sure that you compare corresponding parts. b - 1 24 to triangle PQR. If the perimeter of LMN is 64 units, what is the perimeter of PQR? 3 / . A 108 units C 48 units B 96 units D 36 units 18 2 Read the Item You know the measures of two corresponding sides and the perimeter of LMN. You need to find the perimeter of PQR. Solve the Item 24 4 Triangle LMN ∼ triangle PQR with a scale factor of _ or _ . The ratio 3 18 4 . of the perimeters of LMN to PQR is also _ 3 perimeter of LMN 64 _ _4 ⎫⎬ Scale factor relating LMN to PQR = perimeter of PQR 3 x ⎭ 64 · 3 = 4 · x Find the cross products. 192 = 4x 192 4x _ =_ 4 4 48 = x Multiply. Divide each side by 4. Simplify. The answer is C. e. Square KLMN is similar to square TUVW. If the perimeter of square KLMN is 32 units, what is the perimeter of square TUVW? F 128 units H 64 units G 96 units J 40 units , 8 . / 6 16 5 Lesson 4-7 Similar Polygons 7 8 221 Example 1 (p. 219) Determine whether each pair of polygons is similar. Explain. 1. 2. 5 5 3 18 13 4 6 8 12 7.5 10 6 13.5 8 Example 2 3. In the figure at the right, FGH ∼ KLJ. (p. 220) Write and solve a proportion to find each missing side measure. ' 6 9 - ( 6 y x + 3 , ) Example 3 4. MULTIPLE CHOICE ABC is similar to XYZ. (p. 221) HOMEWORK HELP For Exercises See Examples 5–8 9–12 18, 19 1 2 3 : If the perimeter of ABC is 40 units, what is the perimeter of XYZ? A 10 units C 40 units B 20 units D 80 units # " 9 8 ; $ 16 Determine whether each pair of polygons is similar. Explain. 3 5. 7 6. 4 8 7. 18 16 20 3 3 3 3 5 5 5 5 8. 12 5 15 4 24 8 6 Each pair of polygons is similar. Write and solve a proportion to find each missing side measure. 9. 12 10. x 3 8 8 8 x 5 4 4.8 10 12 11. 29 x 10 21 222 22.4 12. Chapter 4 Proportions and Similarity 14.5 10.5 12.8 12 26 14 7.5 8 x 13. LIFE SCIENCE The scale factor from the model of a human inner ear to the actual ear is 55:2. If one of the bones of the model is 8.25 centimeters long, how long is the actual bone in a human ear? 14. TELEVISION The ratio of the length of EXTRA a wide-screen TV to its width is 16:9. Find the width of a wide-screen TV if the length measures 28 inches. Round to the nearest tenth. PRACTICE See pages 679, 703. H.O.T. Problems 15. CHALLENGE Suppose two rectangles are similar with a scale factor of 2. What is the ratio of their areas? Explain. WR ITING IN MATH Determine whether each statement is always, sometimes, or never true. Explain your reasoning. 16. Any two rectangles are similar. 17. Any two squares are similar. 18. Triangle FGH is similar to triangle RST. G 36 in. F 34 in. 19. Quadrilateral ABCD is similar to quadrilateral WXYZ. R 18 in. " 27 in. H T S ? C 24 inches 2 B 22_ inches 3 1 D 25_ inches 2 # 8 4 in. 9 ; % $ −− What is the length of TS? 1 A 13_ inches 6 in. : If the perimeter of quadrilateral ABCD is 54 units, what is the perimeter of quadrilateral WXYZ? 2 F 13.5 inches H 27 inches G 24 inches J 36 inches 20. ROCK CLIMBING Grace is working her way up a climbing wall. Every 5 minutes she is able to climb 6 feet, but then loses her footing, slips back 1 foot, and decides to rest for 1 minute. If the rock wall is 30 feet tall, how long will it take her to reach the top? Use the draw a diagram strategy. (Lesson 4-6) Solve each proportion. (Lesson 4-5) y 5 21. _ = _ 4 12 22. 120 24 _ =_ b 60 23. 0.6 1.5 _ =_ 5 PREREQUISITE SKILL Graph and connect each pair of ordered pairs. 24. (-2.5, 1.5), (1.5, -3.5) 25. (-2, -1_12 ), (4, 3_12 ) 26. n (Lesson 3-6) (-2_13 , 1), (2, 3_23 ) Lesson 4-7 Similar Polygons 223
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