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. 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. Z W
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.

6. 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

8. You can use Theorem 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. Example 3: Finding Side Lengths in Right Triangles
Find x, y, and z.
Find u, v, and w.

11. 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?

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. 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

14. 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