Slide #1

8.1 Geometric Mean and Pythagorean Theorem

Geometry

Objectives/Assignment

Use Pythagorean theorem to solve problems Use Geometric Mean and Pythagorean Theorem to

solve real-life problems

Geometric Mean

The geometric mean of two positive numbers a and b is the positive number x such that

ax =

xb

If you solve this proportion for x, you find that x = √a ∙ b which is a positive number.

Book Example Pg 397

Geometric Mean Example

For example, the geometric mean of 8 and 18 is 12, because

and also because x = √8 ∙ 18 = x = √144 = 12

812 = 18

12

Practice “Geometric Mean”

Geometric mean

PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.420 mmA3

x

x

210 mm

A4

Solution:

420 mmA3

x

x

210 mm

A4

210

x=

x

420

X2 = 210 ∙ 420

X = √210 ∙ 420

X = 297mm

Write proportion

Cross product property

Simplify

EXAMPLE 1 Find the length of a hypotenuse

Find the length of the hypotenuse of the right triangle.

(hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem

x2 = 62 + 82

x2 = 36 + 64

x2 = 100

x = 10 Find the positive square root.

Substitute.

Square.

Add.

GUIDED PRACTICE for Example 1

Find the unknown side length of the right triangle. Write your answer in simplest radical form.

1.

ANSWER 4

GUIDED PRACTICE for Example 1

Find the unknown side length of the right triangle. 2.

13ANSWER

X = 3.2

Y = X + 5 Y = 3.2 + 5 Y = 8.2

Y² = 32

Y = 5.7

5.7²+ 8² = x²32.49 + 64 = x²96.49 = x²X = 9.8

x² = 20

x = 4.5

4.5²+ 5² = y²20 + 25 = y²45 = y²X = 6.7

4x

x² = 18

x = 4.2 6²+ 4.2² = y²36 + 18 = y²54 = y²X = 7.3

x²+ 5² = 9²x² + 25 = 81x² = 81-25X² = 56X = 7.5

2²+ 2² = x²4 + 4 = x²x² = 8X = 2.8

30²+ 16² = x²900+256 = x²x² = 1156X = 34

x+ 60² = 65²x² +3600 = 4225x² = 4225-3600x² = 625X = 25

?

14² + 48² = 50²

45² + 60² = 80²

50² + 75² = 85²

15² + 36² = 39²