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Use geometric mean to find segment lengths in right triangles.

Objectives. Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems. W. Z. Example 1: Identifying Similar Right Triangles. Write a similarity statement comparing the three triangles.

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Use geometric mean to find segment lengths in right triangles.

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  1. Objectives Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems.

  2. W Z Example 1: Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.

  3. Consider the proportion . In this case, the means of the proportion are the same number, and that number is the geometric mean of the extremes. The geometric meanof two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that , or x2 = ab.

  4. Example 2A: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 4 and 25 Let x be the geometric mean. x2 = (4)(25) = 100 Def. of geometric mean x = 10 Find the positive square root.

  5. Check It Out! Example 2a Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 2 and 8 Let x be the geometric mean. x2 = (2)(8) = 16 Def. of geometric mean x = 4 Find the positive square root.

  6. Check It Out! Example 2b Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 10 and 30 Let x be the geometric mean. x2 = (10)(30) = 300 Def. of geometric mean Find the positive square root.

  7. Example 3: Finding Side Lengths in Right Triangles Find x, y, and z. 62 = (9)(x) 6 is the geometric mean of 9 and x. x = 4 Divide both sides by 9. y is the geometric mean of 4 and 13. y2 = (4)(13) = 52 Find the positive square root. z2 = (9)(13) = 117 z is the geometric mean of 9 and 13. Find the positive square root.

  8. Check It Out! Example 3 Find u, v, and w. 92 = (3)(u) 9 is the geometric mean of u and 3. u = 27 Divide both sides by 3. w2 = (27 + 3)(27) w is the geometric mean of u + 3 and 27. Find the positive square root. v2 = (27 + 3)(3) v is the geometric mean of u + 3 and 3. Find the positive square root.

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