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

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.

Write proportion 210 x = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 X = 297mm
Solution: 210 x Write proportion = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 X = 297mm Cross product property Simplify

Find the positive square root.
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 = Substitute. x2 = Square. x2 = 100 Add. x = 10 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 Find the unknown side length of the right triangle. 2. 13 ANSWER

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

5.7²+ 8² = x² = x² 96.49 = x² X = 9.8 Y² = 32 Y = 5.7

x 4 4.5²+ 5² = y² = y² 45 = y² X = 6.7 x² = 20 x = 4.5

x² = 18 x = 4.2 6²+ 4.2² = y² = y² 54 = y² X = 7.3

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

2²+ 2² = x² 4 + 4 = x² x² = 8 X = 2.8

30²+ 16² = x² = x² x² = 1156 X = 34

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

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