7.2.2 Using Proportional Relationships Key Objectives • Use ratios to make indirect measurements. • Use scale drawings to solve problems. Key Terms • Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. • A scale drawing represents an object as smaller than or larger than its actual size. • A drawing’s scale is the ratio of any length in the drawing to the corresponding actual length. Theorems, Postulates, Corollaries, and Properties • Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a b ,, then the ratio of their perimeters is a b , and the ratio of their areas is a b , or . 2 2 2
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Example 1 Measurement Application
The height of an object is determined from measurements on the ground in this application example. It is given that the student measuring the height is 5 ft 2 inches tall. It is also given that the inflatable ape throws a shadow 10 ft 6 inches long, while the student throws a shadow 3 ft long. Begin by converting all measurements to a single unit, inches. Then, prove that the triangles are similar. AB and DE are parallel because the rays of the Sun that form them are parallel. Therefore, the angles they form with the ground are congruent: ∠ B ≅ ∠ E . Because they are both right angles, ∠ F ≅ ∠ C . Therefore, by Angle-Angle Similarity △ ABC ∼ △ DEF . Set up a proportion with the lengths of the known sides and the unknown length (height of the ape). Substitute the known values for length, cross multiply, and solve for the unknown value. The height of the ape is 217 inches, or 18 ft 1 in.
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