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8.3 Similar Polygons Goals p Identify similar polygons. p Use similar polygons to solve real-life problems. VOCABULARY Similar polygons Similar polygons are two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional. Scale factor The scale factor is the ratio of the lengths of two corresponding sides of two similar polygons. Example 1 Comparing Similar Polygons Decide whether the figures are similar. If they are similar, write a similarity statement. F 8 G K 12 6 L 9 16 12 J N 20 H 15 M Solution As shown, the corresponding angles of FGHJ and KLMN are congruent. Also, the corresponding side lengths are proportional. FG 8 4 KL 6 3 GH 16 4 LM 12 3 20 4 HJ 15 3 MN 12 4 FJ 9 3 KN Answer So, the two figures are similar and you can write FGHJ KLMN . Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 8 • Geometry Notetaking Guide 170 Checkpoint Decide whether the figures are similar. If they are, write the similarity statement. 1. A 8 B D 7 6 12 11 C E 5 F The triangles are not similar. Example 2 Comparing Photographic Enlargements You have a 4-inch by 6-inch photo that you want to use for class election posters. You want the enlargement to be 26 inches wide. How long will it be? 6 4 x Solution Compare the enlargement to the original measurements of the photo. 26 26 in. x in. 4 in. 6 in. 26 x p 6 4 x 39 inches Answer The length of the enlargement will be 39 inches. THEOREM 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. If KLMN PQRS, then P Q K L N M S R KL LM MN NK LM MN NK KL . PQ QR RS SP QR RS SP PQ Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 8 • Geometry Notetaking Guide 171 Example 3 Using Similar Polygons Pentagon CDFGH is similar to pentagon JKLMN. C x D 2 H Find the value of x. K 9 x 9 2 6 x 3 N 6 Solution Set up a proportion that contains CD. CD JK DF KL J G F M L Write proportion. Substitute. Cross multiply and divide by 6 . Checkpoint Complete the following exercises. 2. Verify that these two triangles are similar. Write the similarity statement. Then find the ratio of their perimeters. S T 15 10 6 9 These triangles are similar because the corresponding angles are congruent and the corresponding sides are in the ratio 3 : 2; PQR S VTS; 3 : 2 3. Parallelogram JKLM is similar to parallelogram PQRS. Find the value of x. 8 P R Q V 12 J K 8 M 20 L Q P x 6 S Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 8 • Geometry Notetaking Guide 15 R 172