7.1.2 Ratios in Similar Polygons Key Objectives • Identify similar polygons. • Apply properties of similar polygons to solve problems. Key Terms • Figures that are similar ( ∼ ) have the same shape but not necessarily the same size.
• A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. • Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding sides are proportional. Example 1 Describing Similar Polygons
The pairs of congruent sides and angles in two similar triangles are identified in this example. The lengths of the sides are given. Two pairs of angles are given as congruent. According to the Third Angles Theorem, since ∠ B ≅ ∠ E and ∠ C ≅ ∠ F , then the third angles, ∠ A and ∠ D , are also congruent. To determine whether the sides are congruent, calculate the ratios of the lengths of each hypothesized congruent pair. When setting up the ratios, make sure to put all the lengths from one triangle in the same position, on the top or on the bottom. These ratios are all equal to 2, so all three pairs of sides in the triangles are congruent. In this example, two rectangles are compared to determine whether they are similar. The similarity ratio is calculated. It is given that both figures are rectangles. The lengths of two consecutive sides for each rectangle are given. Since both figures are rectangles, all the angles are right angles and therefore congruent. To determine whether the sides are congruent, calculate the two ratios of side lengths. Both ratios of side lengths are 3/2, so the rectangles are similar (since these are rectangles, the other two sides are congruent pairs with the given sides). The similarity ratio for the two rectangles is the common ratio of the lengths of the sides, 3/2. Rectangle XYZW ∼ rectangle ABCD .
Example 2 Identifying Similar Polygons
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