1. Class Greeting

2. Chapter 8 – Lesson 1
Geometric Mean

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

4. Vocabulary
geometric mean

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

7. 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. Z W
By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.

8. 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.

9. Consider the proportion
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 mean of 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.

10. Another way to think about Geometric Mean
The Geometric mean of a,b is the solution x to the of the proportion

11. Find the geometric mean between 2 and 50.
Let x represent the geometric mean.
Answer: The geometric mean is 10.
Example 1-1a

12. 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
x = 4

13. Find the geometric mean between 25 and 7.
Let x represent the geometric mean.
Answer: The geometric mean is .
Example 1-1b

14. 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.

15. 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.

17. Example 3: Finding Side Lengths in Right Triangles
Find x, y, and z. 13 4
62 = (9)(x)
x = 4
y2 = (4)(13) = 52
z2 = (9)(13) = 117

18. Find c and d in
Example 1-4a

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

20. 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? x
(7.8)2 = 1.6x
x = =
The tree is ≈ 40 m tall.

21. 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? x
(28)2 = 5.5x
x = =
The cliff is  148 ft high.

22. Kahoot!

23. Lesson Summary: Objectives
Use geometric mean to find segment lengths in right triangles.
Apply similarity relationships in right triangles to solve problems.

24. Preview of the Next Lesson:
Objectives
Find the sine, cosine, and tangent of an acute angle.
Use trigonometric ratios to find side lengths in right triangles.

25. Stand Up Please