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Geometry 8.3 Similar Polygons Goals Identify similar polygons Find the ratio of similarity between similar figures. Solve problems involving similar figures. March 23, 2016 Geometry 8.3 Similar Polygons 2 Similar Polygons ABCD and RSTV are similar polygons: Corresponding angles are congruent. Corresponding sides are proportional. 9 A B 6 R 9 C 15 D 6 T 10 8 V 12 March 23, 2016 S Geometry 8.3 Similar Polygons 3 Similar Polygons Corresponding angles are congruent: A R, B S, C T, D V 9 A B 6 R 9 C 15 D 6 T 10 8 V 12 March 23, 2016 S Geometry 8.3 Similar Polygons 4 Similar Polygons Corresponding sides are proportional: 9 A 15 12 9 9 10 8 6 6 3 3 3 3 2 2 2 2 B 6 R 9 C 15 D 6 T 10 8 V 12 March 23, 2016 S Geometry 8.3 Similar Polygons 5 Similar Polygons 9 A Corr. s Sides prop. ABCD ~ RSTV B 6 R 9 C 15 D 6 T 10 8 V 12 March 23, 2016 S Geometry 8.3 Similar Polygons 6 Similar Polygons Corresponding Angles are congruent. Corresponding Sides are proportional. Use the symbol “~” for similar. To show that two polygons are similar, you must prove both things: angles congruent, sides proportional. March 23, 2016 Geometry 8.3 Similar Polygons 7 Similarity Statements List the congruent J Q, K S, L R angles. Write the ratios of the JK JL KL corresponding sides. QS QR SR K S 70 70 J March 23, 2016 L Q Geometry 8.3 Similar Polygons R 8 Example Are these figures similar? Yes Why? Corr. angles congruent Corr. sides proportional. 2 H N Geometry 8.3 Similar Polygons F 1.5 4 6 4 O March 23, 2016 3 E G M 3 8 P 9 Write the similarity statements. E N, F M 2 G P, H O EF FG GH HE = NM MP PO ON EFGH ~ NMPO 3 E H N 1.5 4 6 4 O F G M 3 8 P Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc. March 23, 2016 Geometry 8.3 Similar Polygons 10 Scale Factor If two polygons are similar, the ratio of two corresponding sides is called the scale factor. March 23, 2016 Geometry 8.3 Similar Polygons 11 Scale Factor The scale factor of JKL to QSR is 10/5 or 2/1. The scale factor of QSR to JKL is 5/10 or 1/2. K 10 J March 23, 2016 S 70 5 70 L Q Geometry 8.3 Similar Polygons R 12 Perimeter and Similar Figures 1. ABCD ~ FGHI 2. Find the scale factor of ABCD to FGHI. 3. Find the values of x, y, and z. 4. Find the perimeter of ABCD and FGHI. 5. Find the ratio of the perimeters. A 10 B F 5 G 14 14 x I z H D 4 C March 23, 2016 y Geometry 8.3 Similar Polygons 13 Perimeter and Similar Figures 2. Find the scale factor from ABCD to FGHI. 10 2 5 1 The only known corresponding sides are AB and FG. A 10 B F 5 G 14 14 x I z H D 4 C March 23, 2016 y Geometry 8.3 Similar Polygons 14 Perimeter and Similar Figures 3. Find the values of x, y, and z. 10 14 5 x 10 x 70 2 14 1 y 2 y 14 x7 y7 A 10 2 4 1 z 2z 4 z2 B F 5 G 14 14 x I z H D 4 C March 23, 2016 y Geometry 8.3 Similar Polygons 15 Perimeter and Similar Figures 4. Find the perimeter of ABCD and FGHI. P = 42 A 10 B P = 21 F 5 G 14 14 7 I 2 H D 4 C March 23, 2016 7 Geometry 8.3 Similar Polygons 16 Perimeter and Similar Figures 5. Find the ratio of the perimeters. Ratio of perimeters 2:1 Ratio of Similarity P = 42 A 10 B 2:1 P = 21 F 5 G 14 14 7 I 2 H D 4 C March 23, 2016 7 Geometry 8.3 Similar Polygons 17 Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths and is equal to the similarity ratio. March 23, 2016 Geometry 8.3 Similar Polygons 18 Theorem 8.1 Example 20 These figures are similar. Find the perimeter of the smaller one. 8 P = 100 P=? 20 100 8 P 20 P 800 P 40 March 23, 2016 Geometry 8.3 Similar Polygons 19 Problems to Solve March 23, 2016 Geometry 8.3 Similar Polygons 20 Problem 1 Solve for x and y if the triangles are similar. x+6 20 4 y–2 March 23, 2016 8 6 Geometry 8.3 Similar Polygons 21 Problem 1 Solution Scale Factor is 20/8 x+6 20 8 4 y–2 20Solve x6 8 for x4 8( x 6) 80 x 6 10 x4 March 23, 2016 6 20 Solve y2 8 for y6 8( y 2) 120 y 2 15 y 17 Geometry 8.3 Similar Polygons 22 Problem 2 Find x and y if the figures are similar. x + 10 85 100 32 60 24 95 March 23, 2016 Geometry 8.3 Similar Polygons y 23 Problem 2 Solution x + 10 85 100 60 24 95 Similarity Ratio y = 360 - 100 - 85 - 95 y = 80 March 23, 2016 32 Geometry 8.3 Similar Polygons y 60 x 10 24 32 1920 24 x 240 1680 24 x x 70 24 Problem 3 ABC ~ RST AB = 20 ST = 4 BC = RS Find BC and RS. March 23, 2016 Geometry 8.3 Similar Polygons 25 Problem 3 Solution ABC ~ RST AB = 20 ST = 4 BC = RS Find BC and RS. March 23, 2016 A R 20 x B x Geometry 8.3 Similar Polygons C S 4 T 26 Problem 3 Solution AB BC RS ST 20 x x 4 x 2 80 A R 20 x x 80 16 5 B x4 5 ABC ~ RST x C S 4 T x 8.94 March 23, 2016 Geometry 8.3 Similar Polygons 27 Problem 4 You want to print a picture in your camera. You have two sizes of paper for your printer: 4 × 6 and 5 × 7. Does it matter? Will the pictures printed from each size of paper be similar? 4×6 5×7 March 23, 2016 4 6 5 7 28 30 Geometry 8.3 Similar Polygons Sides not proportional, figures not similar. 28 Problem 5 MNOP has a perimeter of 24. Find the perimeter of QRST if MN = 8 and QR = 12. MN 8 QR 12 8 24 12 P 8 P 288 P 36 March 23, 2016 Geometry 8.3 Similar Polygons 29 Summary Two polygons are similar if they have the same shape, but a different size. If polygons are similar corresponding angles are congruent, and corresponding sides are proportional. The ratio of any two corresponding sides is the scale factor. The ratio of the perimeters is equal to the ratio of two corresponding sides. March 23, 2016 Geometry 8.3 Similar Polygons 30