1. Geometric and Arithmetic Means
Lesson 7-1
Geometric and Arithmetic Means

2. Objectives
Find the arithmetic mean between two numbers (their average)
Find the geometric mean between two numbers
Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse

3. Vocabulary
Arithmetic Mean – between two numbers is their average (add the two numbers and divide by 2)
Geometric Mean – between two numbers is the positive square root of their product

4. Means Arithmetic Mean (AM): is the average of two numbers
example: 4 and AM = (4 + 10)/2 = 14/2 = 7
Geometric Mean (GM): is the square root of their product
example: 4 and GM = √(4•10) = √40 ≈ 6.32
Find the AM and GM of the following numbers
AM GM
a. 6 and 16
b. 4 and 8
c. 5 and 10
d. 2 and 14
(6+16)/2 = 11
√ (6•16) = √96 ≈ 9.8
(4+8)/2 = 6
√ (4•8) = √32 ≈ 5.66
(5+10)/2 = 7.5
√ (5•10) = √50 ≈ 7.07
(2+14)/2 = 8
√ (2•14) = √28 ≈ 5.29

5. Example 1a Find the geometric mean between 2 and 50.
Let x represent the geometric mean.
Definition of geometric mean
Cross products
Take the positive square root of each side.
Simplify.
Answer: The geometric mean is 10.

6. Example 1b a. Find the geometric mean between 3 and 12. Answer: 6
b. Find the geometric mean between 4 and 20.
Answer: 6
Answer: 8.9

7. Application of Geometric Meanb
altitude
hypotenuse (length = a + b)
From similar triangles
a x
--- = ---
x b x
Geometric mean of two numbers a, b is square root of their product, ab
The length of an altitude, x, from the 90° angle to the hypotenuse is the geometric mean of the divided hypotenuse x = ab

8. Example of Geometric Mean6 14 x
altitude
To find x, the altitude to the hypotenuse, we need to find the two pieces the hypotenuse has been divided into: a 6 piece and a 14 piece.
The length of the altitude, x, is the geometric mean of the divided hypotenuse.
x = √6 • 14
= √84
≈ 9.17

9. Example 2a Cross products Take the positive square root of each side.
Use a calculator.
Answer: CD is about 12.7.

10. Example 2b
Answer: about 8.5

11. Example 3
Find c and d in
is the altitude of right triangle JKL. Use Theorem 7.2 to write a proportion.
Cross products
Divide each side by 5.

12. Example 3 cont is the leg of right triangle JKL.
Use the Theorem 7.3 to write a proportion.
Cross products
Take the square root.
Simplify.
Use a calculator.
Answer:

13. Example 4
Find e and f. f
Answer:

14. Summary & Homework Summary: Homework:
The arithmetic mean of two numbers is their average (add and divide by two)
The geometric mean of two numbers is the square root of their product
You can use the geometric mean to find the altitude of a right triangle
Homework:
pg 346, 10, 11, 13-15, 21-24, 29-31