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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. Vocabulary. geometric mean. In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right 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. Vocabulary geometric mean

  3. In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.

  4. Check It Out! Example 1 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, ∆LJK ~ ∆JMK ~ ∆LMJ.

  5. 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.

  6. 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.

  7. 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.

  8. Check It Out! Example 2c Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 8 and 9 Let x be the geometric mean. x2 = (8)(9) = 72 Def. of geometric mean Find the positive square root.

  9. You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means.

  10. Helpful Hint Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers.

  11. 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.

  12. Check It Out! Example 4 A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot?

  13. Check It Out! Example 4 Continued Let x be the height of cliff above eye level. (28)2 = 5.5x 28 is the geometric mean of 5.5 and x. x 142.5 Divide both sides by 5.5. The cliff is about 142.5 + 5.5, or 148 ft high.

  14. Lesson Quiz: Part I Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 8 and 18 2. 6 and 15 12

  15. Lesson Quiz: Part II For Items 3–6, use ∆RST. 3. Write a similarity statement comparing the three triangles. 4. If PS = 6 and PT = 9, find PR. 5. If TP = 24 and PR = 6, find RS. 6. Complete the equation (ST)2 = (TP + PR)(?). ∆RST ~ ∆RPS ~ ∆SPT 4 TP

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