Vocabulary indirect measurement 1

Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures. 2

Additional Example 1: Finding Unknown Lengths in Similar Figures Find the unknown measures in the similar figures. H B 10 cm 31° A y 5.8 cm x 6 cm 11.6 cm 59° J G C 5 cm AB JG = BC HG Write a proportion using corresponding sides. 10 5 6 x = Substitute lengths of the sides. 10 · x = 5 · 6 Find the cross product. 10x = 30 Multiply. 10x 10 30 10 = Divide each side by 12 to isolate the variable. x = 3 HG is 3 centimeters. 3

Additional Example 1 Continued Find the unknown measures in the similar figures. H B 10 cm 31° A y 5.8 cm x 6 cm 11.6 cm 59° J G C 5 cm Step 2 Find y. Corresponding angles of similar triangles have equal angle measures. H corresponds to C y = 59 4

ΔQRS ~ ΔXYZ. Find the unknown measures. Check It Out: Example 1 ΔQRS ~ ΔXYZ. Find the unknown measures. R 12 in. x Z 101° y x 6 in 101° Q S 53° 26° Y 18 in. QR XY = QS YZ X 6 18 12 6 . 18 ; x 9; mz=y=ms = 26°. 5

Additional Example 2: Measurement Application The inside triangle is similar in shape to the outside triangle. Find the length of the base of the inside triangle. Let x = the base of the inside triangle. 8 2 12 x Write a proportion using corresponding side lengths. = 8 · x = 2 · 12 Find the cross products. 8x = 24 Multiply. 8x 8 24 8 = Divide each side by 8 to isolate the variable. x = 3 The base of the inside triangle is 3 inches. 6

Check It Out: Example 2 Van Gogh’s painting Starry Night is 73 cm high by 92 cm long. The museum gift shop sells a postcard of the painting. The image on the postcard is similar to the painting, and it is 10 cm high. What is the length of the postcard? Round to the nearest tenth. 10 73 = x 92 ; 10 92 ≈ 12.6027; 12.6 cm 7

Additional Example 3: Estimating with Indirect Measurement City officials want to know the height of a traffic light. Estimate the height of the traffic light. 27.25 15 48.75 h = Write a proportion. 25 15 50 h Use compatible numbers to estimate. h ft ≈ 5 3 50 h ≈ Simplify. 27.25 ft 5h ≈ 150 Cross multiply. 48.75 ft h ≈ 30 Divide each side by 5 to isolate the variable. The traffic light is about 30 feet tall. 8

Check It Out: Example 3 Gus measures his shadow and the shadow of a lamp post, as shown at right. Estimate the height of the lamp post. 5 • 42 7 = x 5 ; 42 30; 30 ft 9

Find the unknown measures in each pair of similar figures. Lesson Quiz: Part I Find the unknown measures in each pair of similar figures. x = 120 cm y = 98 1. 2. t = 150 cm s = 120 10

Lesson Quiz: Part II 3. The width of the smaller rectangular cake is 5.75 in. The width of a larger rectangular cake is 9.25 in. Estimate the length of the larger rectangular cake. about 15 inches 11