Similar Polygons. Copy both triangles onto tracing paper. Measure and record the sides of each triangle. Cut out both triangles.


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1 7 Similar Polygons MAIN IDEA Identify similar polygons and find missing measures of similar polygons. New Vocabulary polygon similar corresponding parts congruent scale factor Math Online glencoe.com Etra Eamples Personal Tutor SelfCheck Quiz Follow the steps below to discover how the triangles at the right are related. 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.. Epress the ratios _ DF LK, _ EF JK, and _ DE as decimals to the nearest tenth. LJ. 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. The parts of similar figures that match are called corresponding parts. Corresponding Angles A W, B X, C Y, D Z Corresponding Sides AB WX, BC XY, CD YZ, DA ZW 1 Chapter Proportions and Similarity
2 The similar triangles in the Mini Lab suggest the following. Similar Polygons Key Concept Reading Math Congruence The symbol is read is congruent to. Arcs are used to show congruent angles. Words Model Symbols 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. ABC XYZ _ AB_ A X, B Y, C Z, and XY = BC AC_ YZ = XZ Identify Similar Polygons Common Error Do not assume that two polygons are similar just because their corresponding angles are congruent. Their corresponding sides must also be proportional. 1 Determine whether rectangle HJKL is similar to rectangle MNPQ. Eplain. First, check to see if corresponding angles are congruent. Since the two polygons are rectangles, all of their angles are right angles. Therefore, all corresponding angles are congruent. Net, check to see if corresponding sides are proportional. JK_ NP = _ or _ 1 _ HJ MN = _ 7 Since _ 7 and _ 1 _ KL PQ = _ LH_ QM = _ or _ 1 are not equivalent ratios, rectangle HJKL is not similar to rectangle MNPQ. Determine whether these polygons are similar. Eplain. a. b 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 7 Similar Polygons 19
3 Find Missing Measures GEOMETRY Given that polygon WXYZ polygon ABCD, find the missing measure. 1 1 m Reading Math Segment Measure The measure of XY is written as XY. It represents a number. METHOD 1 Write a proportion. The missing measure m is the length of XY. Write a proportion. polygon WXYZ polygon ABCD _ XY BC = _ YZ CD m_ = _ 1 m = 1 m = polygon WXYZ polygon ABCD XY = m, BC =, YZ = 1, and CD =. Find the cross products. Multiply. m = 1 Divide each side by. Scale Factor In Eample, the scale factor from polygon ABCD to polygon WXYZ is _, which means that a length _ on polygon ABCD is as long as a length on polygon WXYZ. METHOD Use the scale factor to write an equation. Find the scale factor from polygon WXYZ to polygon ABCD. scale factor: _ YZ CD = _ 1 or _ The scale factor is the constant of proportionality. _ Words A length on is times as a corresponding length Variable Equation polygon WXYZ m = _ () Write the equation. m = 1 Multiply. Find each missing measure above. long as Let m represent the measure of XY. m = _ on polygon ABCD. c. WZ d. AB Square A square B with a scale factor of :. Notice that the ratio of their perimeters is : or :. m m Square A Perimeter m B m Square A Square B 0 Chapter Proportions and Similarity
4 This and other related eamples suggest the following. Ratios of Similar Figures Key Concept Words If two figures are a_ similar with a scale factor of, then the b perimeters of the figures have a ratio of a_ b. Model a Figure A b Figure B 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. Triangle LMN is similar to triangle PQR. If the perimeter of LMN is units, what is the perimeter of PQR? A units B 9 units 1 C units D units 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 Triangle LMN triangle PQR with a scale factor of _ 1 or _. The ratio of the perimeters of LMN to PQR is also _. perimeter of LMN _ perimeter of PQR = _ Scale factor relating LMN to PQR = Find the cross products. The answer is C. 19 = Multiply. _ 19 _ = Divide each side by. = Simplify. e. Square KLMN is similar to square TUVW. If the perimeter of square KLMN is units, what is the perimeter of square TUVW? 1 F units H units G 9 units J 0 units Lesson 7 Similar Polygons 1
5 Eample 1 (p. 19) Eample (p. 0) Determine whether each pair of polygons is similar. Eplain. 1.. In the figure at the right, FGH KLJ. Write and solve a proportion to find each missing side measure y Eample (p. 1). MULTIPLE CHOICE ABC is similar to XYZ. If the perimeter of ABC is 0 units, what is the perimeter of XYZ? A units B 0 units C 0 units D 0 units 1 For Eercises 9 1, 19 HELP HOMEWORK HELP See Eamples 1 Determine whether each pair of polygons is similar. Eplain Each pair of polygons is similar. Write and solve a proportion to find each missing side measure Chapter Proportions and Similarity
6 1. LIFE SCIENCE The scale factor from the model of a human inner ear to the actual ear is :. If one of the bones of the model is. centimeters long, how long is the actual bone in a human ear? PRACTICE EXTRA PRACTICE See pages 79, TELEVISION The ratio of the length of a widescreen TV to its width is 1:9. Find the width of a widescreen TV if the length measures inches. Round to the nearest tenth. H.O.T. Problems 1. CHALLENGE Suppose two rectangles are similar with a scale factor of. What is the ratio of their areas? Eplain. WRITING IN MATH Determine whether each statement is always, sometimes, or never true. Eplain your reasoning. 1. Any two rectangles are similar. 17. Any two squares are similar. 1. Triangle FGH is similar to triangle RST. F in. in. G 1 in. What is the length of TS? H A 1 _ 1 inches C inches? 7 in. B _ inches D 1 _ inches T R S 19. Quadrilateral ABCD is similar to quadrilateral WXYZ. in. in. If the perimeter of quadrilateral ABCD is units, what is the perimeter of quadrilateral WXYZ? F 1. inches H 7 inches G inches J inches 0. ROCK CLIMBING Grace is working her way up a climbing wall. Every minutes she is able to climb feet, but then loses her footing, slips back 1 foot, and decides to rest for 1 minute. If the rock wall is 0 feet tall, how long will it take her to reach the top? Use the draw a diagram strategy. (Lesson ) Solve each proportion. (Lesson ) 1. _ = _ y. _ 0 b = _ 0. _ 0. = _ 1. n PREREQUISITE SKILL Graph and connect each pair of ordered pairs. (Lesson ). (., 1.), (1., .). (, 1 1_, ) (, _ 1. 1_ ) (, 1 ), (, _ ) Lesson 7 Similar Polygons
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