Lesson 7-1: Geometric Mean 1

Geometric Mean

Lesson 7-1

Lesson 7-1: Geometric Mean 7

a x

x b ab

The geometric mean between two numbers a and b is the

positive number x where . Therefore x = .

Try It: Find the geometric mean of . . .

1. 10 and 40 Answer = 20

2. 1 and 36 Answer = 6

3. 10 and 20 Answer = 14.14

4. 5 and 6 Answer = 5.48

5. 8.1 and 12.2 Answer = 9.94

Lesson 7-1: Geometric Mean 8

CB

A

How does this relate to geometry?

Lesson 7-1: Geometric Mean 9

If the altitude is drawn from the vertex of the 90° angle of a Right ▲ to its hypotenuse, then the two new ▲s formed are similar to the original ▲ and to each other.

Example: ▲ABC ~ ▲DBA ~ ▲DAC

Theorem 7.1

Lesson 7-1: Geometric Mean 10

D CB

A

Lesson 7-1: Geometric Mean 11

The Geometric Means

D CB

A

Recall the three geometric means that you discovered from your Sketchpad activity.

BUT FIRST . . .

Lesson 7-1: Geometric Mean 12

Re-label the Sides (as lengths)

a f

d

ba

c e

fb

a bf

d ec

Lesson 7-1: Geometric Mean 13

a bf

d e

Geometric Mean #1

af

d e

fb

d

f =

f

e

f is the geometric mean of d and e.

What is the proportion that uses f?

Lesson 7-1: Geometric Mean 14

Geometric Mean #2

ba

c e

fb

a bf

d ec

e

b =

b

c

b is the geometric mean of e and c.

What is the proportion that uses b?

Lesson 7-1: Geometric Mean 15

Geometric Mean #3

d

a =

a

c

ba

c

af

d

a bf

d ec

a is the geometric mean of d and c.

What is the proportion that uses a?

Lesson 7-1: Geometric Mean 16

Put them all together

a bf

d ec

d

a =

a

c

e

b =

b

cd

f =

f

e

Lesson 7-1: Geometric Mean 17

The “W” Pattern

a bf

d ecW

Lesson 7-1: Geometric Mean 18

The measure of an altitude drawn from the vertex of the 90° angle of a Right ▲ to its hypotenuse is the geometric mean between the measures (lengths) of the two segments of the hypotenuse.

Example: AD is the geometric mean of BD and DC.

Theorem 7.2

Lesson 7-1: Geometric Mean 19

Try it !

a bf

d ec

Given: d = 4 and e = 10

Find:a = ___b = ___c = ___f = ___

Lesson 7-1: Geometric Mean 20

Solution:

4

f =

f

10

4

a =

a

1410

b =

b

14

a bf

4 1014

a = 7.48 f = 6.32 b = 11.83

Proportions

Answers