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5-7 Indirect Measurement Warm Up Problem of the Day Lesson Presentation Course 3 Warm Up Solve each proportion. 1. 3 = x 5 75 3. 9 = x 27 6 x = 45 x=2 2. 6 = 2.4 x 8 4. x = 8 3.5 7 x = 20 x=4 Problem of the Day A plane figure is dilated and gets 50% larger. What scale factor should you use to dilate the figure back to its original size? (Hint: The answer is not 12.) 2 3 Learn to find measures indirectly by applying the properties of similar figures. Vocabulary indirect measurement Sometimes, distances cannot be measured directly. One way to find such a distance is to use indirect measurement, a way of using similar figures and proportions to find a measure. Additional Example 1: Geography Application Triangles ABC and EFG are similar. Find the length of side EG. F B 9 ft 3 ft A 4 ft C E x Triangles ABC and EFG are similar. G Additional Example 1 Continued Triangles ABC and EFG are similar. Find the length of side EG. AB = EF AC EG Set up a proportion. 3 9 = 4 x Substitute 3 for AB, 4 for AC, and 9 for EF. 3x = 36 Find the cross products. 3x = 36 Divide both sides by 3. 3 3 x = 12 The length of side EG is 12 ft. Check It Out: Example 1 Triangles DEF and GHI are similar. Find the length of side HI. H E 2 in D x 8 in 7 in F G Triangles DEF and GHI are similar. I Check It Out: Example 1 Continued Triangles DEF and GHI are similar. Find the length of side HI. DE = GH EF HI 2 8 = 7 x 2x = 56 Set up a proportion. Substitute 2 for DE, 7 for EF, and 8 for GH. Find the cross products. 2x = 56 Divide both sides by 2. 2 2 x = 28 The length of side HI is 28 in. Additional Example 2: Problem Solving Application A 30-ft building casts a shadow that is 75 ft long. A nearby tree casts a shadow that is 35 ft long. How tall is the tree? 1 Understand the Problem The answer is the height of the tree. List the important information: • The length of the building’s shadow is 75 ft. • The height of the building is 30 ft. • The length of the tree’s shadow is 35 ft. Additional Example 2 Continued 2 Make a Plan Use the information to draw a diagram. h 30 feet 35 feet 3 75 feet Solve Draw dashed lines to form triangles. The building with its shadow and the tree with its shadow form similar right triangles. Additional Example 2 Continued 3 Solve 30 = h 75 35 Corresponding sides of similar figures are proportional. 75h = 1050 Find the cross products. 75h = 1050 75 75 Divide both sides by 75. h = 14 The height of the tree is 14 feet. Additional Example 2 Continued 4 Look Back 75 Since 30 = 2.5, the building’s shadow is 2.5 times its height. So, the tree’s shadow should also be 2.5 times its height and 2.5 of 14 is 35 feet. Check It Out: Example 2 A 24-ft building casts a shadow that is 8 ft long. A nearby tree casts a shadow that is 3 ft long. How tall is the tree? 1 Understand the Problem The answer is the height of the tree. List the important information: • The length of the building’s shadow is 8 ft. • The height of the building is 24 ft. • The length of the tree’s shadow is 3 ft. Check It Out: Example 2 Continued 2 Make a Plan Use the information to draw a diagram. h 24 feet 3 feet 3 8 feet Solve Draw dashed lines to form triangles. The building with its shadow and the tree with its shadow form similar right triangles. Check It Out: Example 2 Continued 3 Solve 24 = h 8 3 Corresponding sides of similar figures are proportional. 72 = 8h Find the cross products. 72 = 8h 8 8 Divide both sides by 8. 9=h The height of the tree is 9 feet. Check It Out: Example 2 Continued 4 Look Back 8 1 Since 24 = 3 , the building’s shadow is 1 3 times its height. So, the tree’s shadow should also be 1 times its 3 1 height and 3 of 9 is 3 feet. Lesson Quiz 1. Vilma wants to know how wide the river near her house is. She drew a diagram and labeled it with her measurements. How wide is the river? 7.98 m w 5m 7m 5.7 m 2. A yardstick casts a 2-ft shadow. At the same time, a tree casts a shadow that is 6 ft long. How tall is the tree? 9 ft