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

Book Example Pg 397

Geometric Mean Example

 For example, the geometric mean of 8 and 18 is 12 , because 8 12 = 12 18 and also because x = √8 ∙ 18 = x = √144 = 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.

A4 210 mm x A3 420 mm x

A4 x 210 mm Solution: 210 x x = 420 X 2 = 210 ∙ 420 X = √210 ∙ 420 A3 420 mm x Write proportion Cross product property Simplify

X = 297mm

EXAMPLE 1 Find the length of a hypotenuse Find the length of the hypotenuse of the right triangle.

( hypotenuse )

2 = (

leg

) 2 + (

leg

) 2

x

2 = 6 2 + 8 2

x

2 = 36 + 64

x

2 = 100

x

= 10

Pythagorean Theorem Substitute.

Square.

Add.

Find the positive square root.

GUIDED PRACTICE for Example 1

1.

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

ANSWER 4

GUIDED PRACTICE for Example 1

2.

Find the unknown side length of the right triangle. ANSWER 13

X = 3.2

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

Y² = 32 Y = 5.7

5.7²+ 8² = x² 32.49 + 64 = x² 96.49 = x² X = 9.8

x 4

x ² = 20 x = 4.5

4.5²+ 5² = y² 20 + 25 = y² 45 = y² X = 6.7

x ² = 18 x = 4.2

6 ²+ 4.2² = y² 36 + 18 = y² 54 = y² X = 7.3

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

2 ²+ 2² = x² 4 + 4 = x² x² = 8 X = 2.8

30²+ 16² = x² 900+256 = x² x² = 1156 X = 34

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

14² + 48² = 50² 15² + 36² = 39² 50² + 75² = 85² 45² + 60² = 80²

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