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D. N. A. 1) Find the ratio of BC to DG. A B C D E F G 0 10 20 30 2) Solve each proportion. 6 5 a) x 7 2x 3 4 b) 5x 3 Slides Skills Practice Practice 1, 4, 5 1-2 1-2 17, 21-27 3-6 3-6 Similar Polygons Chapter 7-2 • Identify similar figures. • Solve problems involving scale factors. • similar polygons • scale factor Standard 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. Lesson 2 MI/Vocab Similar Polygons • Have congruent corresponding angles. • Have proportional corresponding sides. ABCD ~ EFGH • “~” means “is similar to” A E E A F H G D B C B F C G D H AB BC CD DA EF FG GH HE Writing Similarity Statements • Decide if the polygons are similar. If they are, write a similarity statement. 6 AB 6 3 A A W WY 4 2 B Y 15 BC 9 3 C Z YZ 6 2 D X 12 CD 12 3 All corr. sides are D 4 ZX 8 2 proportionate and all W DA 15 3 corr. angles are 10 XW 10 2 ABCD ~ WYZX X 8 B 9 C Y 6 Z Scale Factor • The ratio of the lengths of two corresponding sides. • In the previous example the scale factor is 3:2. Similar Polygons A. Determine whether each pair of figures is similar. Justify your answer. The vertex angles are marked as 40º and 50º, so they are not congruent. Lesson 2 Ex1 Similar Polygons Since both triangles are isosceles, the base angles in each triangle are congruent. In the first triangle, the base angles measure and in the second triangle, the base angles measure Answer: None of the corresponding angles are congruent, so the triangles are not similar. Similar Polygons B. Determine whether each pair of figures is similar. Justify your answer. All the corresponding angles are congruent. Similar Polygons Now determine whether corresponding sides are proportional. The ratios of the measures of the corresponding sides are equal. Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so ΔABC ~ ΔRST. A. Determine whether the pair of figures is similar. A. Yes, ΔAXE ~ ΔWRT. B. Yes, ΔAXE ~ ΔRWT. C. No, the Δ's are not ~. D. not enough information B. Determine whether the pair of figures is similar. A. Yes, ΔTRS ~ ΔNGA. B. Yes, ΔTRS ~ ΔGNA. C. No, the Δ's are not ~. D. not enough information ARCHITECTURE An architect prepared a 12-inch model of a skyscraper to look like a real 1100-foot building. What is the scale factor of the model compared to the real building? Before finding the scale factor you must make sure that both measurements use the same unit of measure. 1 foot = 12 inches Answer: The ratio comparing the two heights is or 1:1100. The scale factor is means that the model is , which the height of the real skyscraper. Animation: Similar Polygons Each pair of polygons is similar. Find x and y. 1) 12 10 x 2) 4 .5 9 y 8 y 10 2 .5 x 3) 5 36 x 40 y 18 30 A space shuttle is about 122 feet in length. The Science Club plans to make a model of the space shuttle with a length of 24 inches. What is the scale factor of the model compared to the real space shuttle? A. B. C. D. Proportional Parts and Scale Factor A. The two polygons are similar. Write a similarity statement. Then find x, y, and UV. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV Proportional Parts and Scale Factor Now write proportions to find x and y. To find x: Similarity proportion Cross products Multiply. Divide each side by 4. Proportional Parts and Scale Factor To find y: Similarity proportion AB = 6, RS = 4, DE = 8, UV = y + 1 Cross products Multiply. Subtract 6 from each side. Divide each side by 6 and simplify. Proportional Parts and Scale Factor Proportional Parts and Scale Factor B. The two polygons are similar. Find the scale factor of polygon ABCDE to polygon RSTUV. The scale factor is the ratio of the lengths of any two corresponding sides. Answer: A. The two polygons are similar. Write a similarity statement. A. TRAP ~ OZDL B. TRAP ~ OLDZ C. TRAP ~ ZDLO D. TRAP ~ ZOLD B. The two polygons are similar. Solve for a. A. a = 1.4 B. a = 3.75 C. a = 2.4 D. a = 2 C. The two polygons are similar. Solve for b. A. b = 7.2 B. b = 1.2 C. D. b = 7.2 D. The two polygons are similar. Solve for ZO. A. 7.2 B. 1.2 C. 2.4 D. E. The two polygons are similar. What is the scale factor of polygon TRAP to polygon ZOLD? A. B. C. D. 1. 2. 3. 4. A B C D Enlargement or Reduction of a Figure Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5. If the length and width of PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ? Write proportions for finding side measures. Let one long side of each WXYZ and PQRS be and one short side of each WXYZ and PQRS be Enlargement or Reduction of a Figure WXYZ PQRS WXYZ PQRS Quadrilateral GCDE is similar to quadrilateral JKLM with a scale factor of . If two of the sides of GCDE measure 7 inches and 14 inches, what are the lengths of the corresponding sides of JKLM? A. 9.8 in, 19.6 in B. 7 in, 14 in C. 6 in, 12 in D. 5 in, 10 in Scales on Maps The scale on the map of a city is inch equals 2 miles. On the map, the width of the city at its widest point is inches. The city hosts a bicycle race across town at its widest point. Tashawna bikes at 10 miles per hour. How long will it take her to complete the race? Explore Every equals 2 miles. The distance across the city at its widest point is Scales on Maps Plan Create a proportion relating the measurements to the scale to find the distance in miles. Then use the formula to find the time. Solve Cross products Divide each side by 0.25. The distance across the city is 30 miles. Scales on Maps Divide each side by 10. It would take Tashawna 3 hours to bike across town. Examine To determine whether the answer is reasonable, reexamine the scale. If 0.25 inches = 2 miles, then 4 inches = 32 miles. The distance across the city is approximately 32 miles. At 10 miles per hour, the ride would take about 3 hours. The answer is reasonable. Answer: 3 hours An historic train ride is planned between two landmarks on the Lewis and Clark Trail. The scale on a map that includes the two landmarks is 3 centimeters = 125 miles. The distance between the two landmarks on the map is 1.5 centimeters. If the train travels at an average rate of 50 miles per hour, how long will the trip between the landmarks take? A. 3.75 hr B. 1.25 hr C. 5 hr D. 2.5 hr Forced Perspective Using Ratios Example #1 • The Perimeter of a rectangle is 60 cm. The ratio of AB:BC is 3:2. Find the length and width of the A B rectangle. 3:2 is in lowest terms. D AB:BC could be 3:2, 6:4, 9:6, 12:8, etc. AB = 3x Perimeter = l + w+ l + w BC = 2x 60 = 3x + 2x + 3x + 2x 60 = 10x L = 3(6) = 18 x=6 W = 2(6) = 12 C Find the measures of the sides of each triangle. 12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. 13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters. 14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet. Find the measures of the angles in each triangle. 15) The ratio of the measures of the angles is 4:5:6. Using Ratios Example #2 • The angle measures in ABC are in the extended ratio of 2:3:4. Find the measure of the three angles. mA+ mB+ mC = 180o Triangle Sum Thm. 2x + 3x + 4x = 180o 9x = 180o B 4x x = 20o mA = 40o mB = 60o mC = 80o A 2x 3x C