Lesson 7-1

Geometric and Arithmetic Means

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

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

MeansArithmetic Mean (AM): is the average of two numbers example: 4 and 10 AM = (4 + 10)/2 = 14/2 = 7

Geometric Mean (GM): is the square root of their product example: 4 and 10 GM = √(4•10) = √40 = 2√10

Find the AM and GM of the following numbers AM GMa. 6 and 16

b. 4 and 8

c. 5 and 10

d. 2 and 14

(6+16)/2 = 11 √ (6•16) = √96 =

(4+8)/2 = 6

(5+10)/2 = 7.5

(2+14)/2 = 8

√ (4•8) = √32 =

√ (5•10) = √50 =

√ (2•14) = √28 =

4√6

4√2

5√2

2√7

Example 1

Find the geometric mean between 2 and 50.

Definition of geometric mean

Let x represent the geometric mean.

Cross products

Take the positive square root of each side.

Simplify.

Answer: The geometric mean is 10.

Examples 2 & 3

a. Find the geometric mean between 3 and 12.

b. Find the geometric mean between 4 and 20.

Answer: 6

Answer: √80= 4√5

Application of Geometric Mean

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

a

baltitude

hypotenuse (length = a + b)

a x--- = --- x b

From similar triangles

x

Example 4 of Geometric Mean

altitude

6

14x

=x = √6 • 14 = √84

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.

2√21

Example 5

Cross products

Take the positive square root of each side.

Example 6

Answer: 6√2

Example 7

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.

is the leg of right triangle JKL.

Use the Theorem 7.3 to write a proportion.

Cross products

Take the square root.

Simplify.

Example 7 cont

Find e and f.

f

Example 8

Answer: f= 4√5, e= 8√5

Summary & Homework

• Summary:– 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 (13-16, 22-32 even)