Evaluating Algebraic Expressions 5-6 Indirect Measurement Extension of MG1.2 Construct and read drawings and models made to scale. California Standards

Evaluating Algebraic Expressions 5-6 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.

Evaluating Algebraic Expressions 5-6 Indirect Measurement Additional Example 1: Geography Application Triangles ABC and EFG are similar. Triangles ABC and EFG are similar. Find the length of side EG. B AC 3 ft 4 ft F E G 9 ft x

Evaluating Algebraic Expressions 5-6 Indirect Measurement Additional Example 1 Continued AB AC = EF EG Set up a proportion. Substitute 3 for AB, 4 for AC, and 9 for EF. 3x = 36Find the cross products. The length of side EG is 12 ft. x = 12 Triangles ABC and EFG are similar. Find the length of side EG. 3 4 = 9 x 3x3x 3 = 36 3 Divide both sides by 3.

Evaluating Algebraic Expressions 5-6 Indirect Measurement Check It Out! Example 1 Triangles DEF and GHI are similar. Triangles DEF and GHI are similar. Find the length of side HI. 2 in E DF 7 in H G I 8 in x

Evaluating Algebraic Expressions 5-6 Indirect Measurement Check It Out! Example 1 Continued DE EF = GH HI Set up a proportion. Substitute 2 for DE, 7 for EF, and 8 for GH. 2x = 56Find the cross products. The length of side HI is 28 in. x = = 8 x 2x 2 = 56 2 Divide both sides by 2. Triangles DEF and GHI are similar. Find the length of side HI.

Evaluating Algebraic Expressions 5-6 Indirect Measurement 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? Additional Example 2: Problem Solving Application 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.

Evaluating Algebraic Expressions 5-6 Indirect Measurement Additional Example 2 Continued Use the information to draw a diagram. 2 Make a Plan Draw dashed lines to form triangles. The building with its shadow and the tree with its shadow form similar right triangles. Solve 3 h 35 feet 75 feet 30 feet

Evaluating Algebraic Expressions 5-6 Indirect Measurement = h 35 Solve 3 Corresponding sides of similar figures are proportional. 75h = 1050Find the cross products. The height of the tree is 14 feet. h = 14 75h 75 = Divide both sides by 75. Additional Example 2 Continued

Evaluating Algebraic Expressions 5-6 Indirect Measurement Since = 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. 4 Look Back Additional Example 2 Continued

Evaluating Algebraic Expressions 5-6 Indirect Measurement 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? Check It Out! Example 2 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.

Evaluating Algebraic Expressions 5-6 Indirect Measurement Use the information to draw a diagram. 2 Make a Plan Draw dashed lines to form triangles. The building with its shadow and the tree with its shadow form similar right triangles. Solve 3 Check It Out! Example 2 Continued h 3 feet 8 feet 24 feet

Evaluating Algebraic Expressions 5-6 Indirect Measurement 24 8 = h3 h3 Solve 3 Corresponding sides of similar figures are proportional. 72 = 8hFind the cross products. The height of the tree is 9 feet. 9 = h 72 8 = 8h 8 Divide both sides by 8. Check It Out! Example 2 Continued

Evaluating Algebraic Expressions 5-6 Indirect Measurement Since =, the building’s shadow is times its height. So, the tree’s shadow should also be times its height and of 9 is 3 feet. 4 Look Back Check It Out! Example 2 Continued