This document provides examples and explanations of similar figures and proportions. It includes: - Three examples of using proportions to find missing side lengths in similar figures. The side lengths scale proportionally between the similar figures. - A lesson quiz with three questions testing the concept of similar figures and using proportions to find missing measures. - Explanations of key vocabulary like corresponding sides and angles, and notation for showing similar figures.

1. Warm Up California Standards Lesson Presentation Preview

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

3. Preparation for MG1.2 Construct and read drawings and models made to scale. California Standards

4. Vocabulary similar corresponding sides corresponding angles

5. 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.

6. 43° 43°

7.  A is read as “angle A .” ∆ ABC is read as “triangle ABC .” “∆ ABC ~∆ EFG ” is read as “triangle ABC is similar to triangle EFG .” Reading Math

8. 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.

9. Additional Example 1 Continued 50  48 The ratios are equal. Rectangle J is similar to rectangle K . The notation J ~ K shows similarity. The ratios are not equal. Rectangle J is not similar to rectangle L . Therefore, rectangle K is not similar to rectangle L . 20 = 20 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 ? = length of rectangle J length of rectangle L width of rectangle J width of rectangle L 10 12 4 5 ? =

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

11. Check It Out! Example 1 Continued 16  20 The ratios are equal. Rectangle A is similar to rectangle B . The notation A ~ B shows similarity. The ratios are not equal. Rectangle A is not similar to rectangle C . Therefore, rectangle B is not similar to rectangle C . 24 = 24 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 ? = length of rectangle A length of rectangle C width of rectangle A width of rectangle C 8 5 4 2 ? =

12. 14 ∙ 1.5 = w ∙ 10 Find the cross products. Divide both sides by 10. 2.1 = w Set up a proportion. Let w be the width of the picture on the Web page. The picture on the Web page should be 2.1 in. wide. 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? Additional Example 2: Finding Missing Measures in Similar Figures 21 = 10 w 10 w 10 21 10 = width of a picture width on Web page height of picture height on Web page 14 w = 10 1.5

13. 56 ∙ 10 = w ∙ 40 Find the cross products. Divide both sides by 40. 14 = w Set up a proportion. Let w be the width of the painting on the Poster. The painting displayed on the poster should be 14 in. long. Check It Out! Example 2 560 = 40 w 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? 40 w 40 560 40 = width of a painting width of poster length of painting length of poster 56 w = 40 10

14. Additional Example 3: Business Application 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? Set up a proportion. 4.5 • x = 3 • 6 Find the cross products. 4.5 x = 18 Multiply. 3 ft x ft 4.5 in. 6 in. = side of small triangle base of small triangle side of large triangle base of large triangle =

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

16. Check It Out! Example 3 Set up a proportion. 18 ft • x in. = 24 ft • 4 in. Find the cross products. 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 x = 96 Multiply. 24 ft x in. 18 ft 4 in. = side of large triangle side of small triangle base of large triangle base of small triangle =

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

18. Lesson Quiz Use the properties of similar figures to answer each question. 1. Which rectangles are similar? A and B are similar. 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. 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.