Key Objectives • Use geometric mean to find segment lengths in right triangles. • Apply similarity relationships in right triangles to solve problems. Key Terms • The geometric mean of two positive numbers is the positive square root of their product. Theorems, Postulates, Corollaries, and Properties • Theorem The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. • Corollary The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. • Corollary The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Example 1 Identifying Similar Right Triangles 8.1.1 Similarity in Right Triangles
The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the other triangle. The altitude to △ ABC forms a right angle with the hypotenuse. The altitude YW forms two smaller triangles, △ XWY and △ YWZ , from the orginal triangle, △ XYZ . The three triangles are similar. This similarity is expressed as △ XYZ ∼ △ YWZ ∼ △ XWY .
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