Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Warm Up Solve each proportion. = = 1. b = 10 2. y = 8 = = 3. 4. m = 52 30 3 9 = 56 35 y 5 = 1. b = 10 2. y = 8 4 12 p 9 = 56 m 28 26 = 3. 4. m = 52 p = 3

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.

Learn to determine whether figures are similar and to find missing dimensions in similar figures.

Vocabulary similar corresponding sides corresponding angles

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.

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.

Additional Example 1 Continued Compare the ratios of corresponding sides to see if they are equal. length of rectangle J length of rectangle K width of rectangle J width of rectangle K 10 5 4 2 ? = 20 = 20 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 width of rectangle J width of rectangle L 10 12 4 5 ? = 50  48 The ratios are not equal. Rectangle J is not similar to rectangle L. Therefore, rectangle K is not similar to rectangle L.

Which rectangles are similar? Check It Out: Example 1 Which rectangles are similar? 8 ft A 6 ft B C 5 ft 4 ft 3 ft 2 ft Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

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 width of rectangle A width of rectangle B 8 6 4 3 ? = 24 = 24 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 width of rectangle A width of rectangle C 8 5 4 2 ? = 16  20 The ratios are not equal. Rectangle A is not similar to rectangle C. Therefore, rectangle B is not similar to rectangle C.

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 height of picture height on Web page 14 w = 10 1.5 14 ∙ 1.5 = w ∙ 10 Find the cross products. 21 = 10w 10w 10 21 = Divide both sides by 10. 2.1 = w The picture on the Web page should be 2.1 in. wide.

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. width of a painting width of poster length of painting length of poster 56 w = 40 10 56 ∙ 10 = w ∙ 40 Find the cross products. 560 = 40w 40w 40 560 = Divide both sides by 40. 14 = w The painting displayed on the poster should be 14 in. long.

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? 6 in. x ft 4.5 in. 3 ft = Set up a proportion. 4.5 • x = 3 • 6 Find the cross products.

Additional Example 3 Continued Multiply. x = = 4 18 4.5 Solve for x. The third side of the triangle is 4 ft long.

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? 24 ft x in. 18 ft 4 in. = Set up a proportion. 18 • x = 24 • 4 Find the cross products.

Check It Out: Example 3 Continued Multiply. x =  5.3 96 18 Solve for x. The third side of the triangle is about 5.3 in. long.

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 18 18

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.

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. 20 20

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 21 21

Lesson Quiz for Student Response Systems 3. Which triangles are similar? A. X and Y B. X and Z C. Y and Z D. none X Y Z 22 22