1. Objectives
Students will learn how to use geometric mean to find segment lengths in right triangles and apply similarity relationships in right triangles to solve problems.

2. Vocabulary
geometric mean

3. The geometric mean of two positive numbers is the positive square root of their product.
Consider the proportion In this case, the means of the proportion are the same number, and that number (x) is the geometric mean of the extremes.

4. Example 1A: Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
4 and 9
Let x be the geometric mean.
Def. of geometric mean
Cross multiply
x = 6
Find the positive square root.

5. Example 1b: Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
6 and 15
Let x be the geometric mean.
Def. of geometric mean
Cross multiply
Find the positive square root.

6. Example 1c: Finding Geometric Means
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.
Def. of geometric mean
Cross multiply
Find the positive square root.

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

8. Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers.
Helpful Hint
It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Example

9. Use Pythagorean Theorem to find c:

10. Use Pythagorean Theorem to find x:

11. Use Pythagorean Theorem to find y:

12. A 10 ladder leans against a wall. It is 5 from the base of the wall
A 10 ladder leans against a wall. It is 5 from the base of the wall. How high does the ladder reach?
The Height is 8m

13. I Do: Watch!! m = 2, n = 10, h = ___ 2 2 10 10 c a b n m h c a b n m h

14. Whiteboard m = 2, n = 10, b = ___ 2 2 10 10 12 c a b n m h c a b n m h

15. Whiteboard n = 27, c = 30, b = ___ 27 27 30 c a b n m h a m h c a b n