8.1.1 Similarity in Right Triangles (continued)
Example 2 Finding Geometric Means
The geometric means of pairs of numbers are calculated in these examples. The geometric mean of two numbers is the positive square root of their product.
Example 3 Finding Side Lengths in Right Triangles
There are two corollaries that describe length relationships in triangles formed by dropping an altitude line in a right triangle. 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. 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. The lengths of the altitude and two sides of a right triangle are calculated using the geometric mean corollaries. The lengths of the two segments of the hypotenuse formed by the altitude are given. To find the value of y , set up a proportion using ratios involving y and a known side in two of the similar triangles. For example, the ratio of y to the hypotenuse of the original triangle is y /(8 + 3) = y /11. Cross multiply and solve for the value of y . The solution yields the positive square root of 33 as the value of y . Use similar proportions to find that = = x z 2 6 and 2 22.
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