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Understanding Similarity in Right Triangles and Geometric Means

Learn about altitude-to-hypotenuse relationships, geometric and arithmetic means, and how to find side lengths and geometric means in right triangles. Practice examples and quizzes provided.

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Understanding Similarity in Right Triangles and Geometric Means

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  1. 9.3 Similarity in Right Triangles Altitude-to-Hypotenuse Geometric Mean Arithmetic Mean

  2. By AA~ the big triangle is similar to each of the little triangles and by the transitive property they are all similar.

  3. W Z Example 1A: 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.

  4. Check It Out! Example 1B Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions.

  5. Example 2A/B: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 4 and 25 5 and 30

  6. 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.

  7. Example 3: Finding Side Lengths in Right Triangles Find x, y, and z. Find u, v, and w.

  8. Example 4: Measurement Application To estimate the height of a Douglas fir, Jan positions herself so that her lines of sight to the top and bottom of the tree form a 90º angle. Her eyes are about 1.6 m above the ground, and she is standing 7.8 m from the tree. What is the height of the tree to the nearest meter?

  9. 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?

  10. 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

  11. 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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