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Similar Figures Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Similar Figures Warm Up Solve each proportion. 1. 3 = b 9 30 3. p = 4 9 12 b = 10 y = 56 2. 5 35 y=8 p=3 4. 28 = 56 m = 52 26 m Similar Figures Problem of the Day A rectangle that is 10 in. wide and 8 in. long is the same shape as one that is 8 in. wide and x in. long. What is the length of the smaller rectangle? 6.4 in. Similar Figures Learn to determine whether figures are similar and to find missing dimensions in similar figures. Similar Figures Vocabulary similar corresponding sides corresponding angles Similar Figures Similar figures have the same shape, but not necessarily the same size. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if the lengths of corresponding sides are proportional and the corresponding angles have equal measures. Similar Figures Similar Figures Additional Example 1: Identifying Similar Figures Which rectangles are similar? Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent. Similar Figures Additional Example 1 Continued Compare the ratios of corresponding sides to see if they are equal. length of rectangle J length of rectangle K 10 ? 4 5 =2 20 = 20 width of rectangle J width of rectangle K The ratios are equal. Rectangle J is similar to rectangle K. The notation J ~ K shows similarity. length of rectangle J length of rectangle L 10 ? 4 12 = 5 width of rectangle J width of rectangle L 50 48 The ratios are not equal. Rectangle J is not similar to rectangle L. Therefore, rectangle K is not similar to rectangle L. Similar Figures Check It Out: Example 1 Which rectangles are similar? 8 ft A 4 ft 6 ft B 3 ft 5 ft C 2 ft Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent. Similar Figures Check It Out: Example 1 Continued Compare the ratios of corresponding sides to see if they are equal. length of rectangle A length of rectangle B 8 ? 4 6 =3 24 = 24 width of rectangle A width of rectangle B The ratios are equal. Rectangle A is similar to rectangle B. The notation A ~ B shows similarity. length of rectangle A length of rectangle C 8 ? 4 5= 2 width of rectangle A width of rectangle C 16 20 The ratios are not equal. Rectangle A is not similar to rectangle C. Therefore, rectangle B is not similar to rectangle C. Similar Figures Additional Example 2: Finding Missing Measures in Similar Figures A picture 10 in. tall and 14 in. wide is to be scaled to 1.5 in. tall to be displayed on a Web page. How wide should the picture be on the Web page for the two pictures to be similar? Set up a proportion. Let w be the width of the picture on the Web page. width of a picture width on Web page 14 10 = w 1.5 height of picture height on Web page 14 ∙ 1.5 = w ∙ 10 Find the cross products. 21 = 10w 21 10w Divide both sides by 10. = 10 10 2.1 = w The picture on the Web page should be 2.1 in. wide. Similar Figures Check It Out: Example 2 A painting 40 in. long and 56 in. wide is to be scaled to 10 in. long to be displayed on a poster. How wide should the painting be on the poster for the two pictures to be similar? Set up a proportion. Let w be the width of the painting on the Poster. 56 40 width of a painting length of painting = w 10 width of poster length of poster 56 ∙ 10 = w ∙ 40 Find the cross products. 560 = 40w 560 40w = 40 40 14 = w Divide both sides by 40. The painting displayed on the poster should be 14 in. long. Similar Figures Additional Example 3: Using Equivalent Ratios to Find Missing Dimensions A T-shirt design includes an isosceles triangle with side lengths 4.5 in, 4.5 in., and 6 in. An advertisement shows an enlarged version of the triangle with two sides that are each 3 ft. long. What is the length of the third side of the triangle in the advertisement? 4.5 in. = 6 in. 3 ft x ft Set up a proportion. 4.5 • x = 3 • 6 Find the cross products. Similar Figures Additional Example 3 Continued 4.5x = 18 x = 18 = 4 4.5 Multiply. Solve for x. The third side of the triangle is 4 ft long. Similar Figures Check It Out: Example 3 A flag in the shape of an isosceles triangle with side lengths 18 ft, 18 ft, and 24 ft is hanging on a pole outside a campground. A camp t-shirt shows a smaller version of the triangle with two sides that are each 4 in. long. What is the length of the third side of the triangle on the t-shirt? 18 ft = 24 ft 4 in. x in. Set up a proportion. 18 • x = 24 • 4 Find the cross products. Similar Figures Check It Out: Example 3 Continued 18x = 96 x = 96 5.3 18 Multiply. Solve for x. The third side of the triangle is about 5.3 in. long. Similar Figures Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems Similar Figures Lesson Quiz Use the properties of similar figures to answer each question. 1. A rectangular house is 32 ft wide and 68 ft long. On a blueprint, the width is 8 in. Find the length on the blueprint. 17 in. 2. Karen enlarged a 3 in. wide by 5 in. tall photo into a poster. If the poster is 2.25 ft wide, how tall is it? 3.75 ft 3. Which rectangles are similar? A and B are similar. Similar Figures Lesson Quiz for Student Response Systems 1. A rectangular garden is 45 ft wide and 70 ft long. On a blueprint, the width is 9 in. Identify the length on the blueprint. A. 7 in. B. 9 in. C. 12 in. D. 14 in. Similar Figures Lesson Quiz for Student Response Systems 2. Rob enlarged a 4 in. wide by 7 in. tall picture into a banner. If the banner is 3.5 ft wide, how tall is it? A. 6.05 ft B. 6.125 ft C. 7 ft D. 7.125 ft Similar Figures Lesson Quiz for Student Response Systems 3. Which triangles are similar? A. X and Y Y B. X and Z C. Y and Z D. none X Z