1. Slide #1

2. 8.1 Geometric Mean and Pythagorean Theorem
Geometry

3. Objectives/Assignment
Use Pythagorean theorem to solve problems
Use Geometric Mean and Pythagorean Theorem to solve real-life problems

4. 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

5. Book Example Pg 397

6. 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

7. Practice “Geometric Mean”

8. 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.

9. 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

11. 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.

12. GUIDED PRACTICE
for Example 1 1.
Find the unknown side length of the right triangle. Write your answer in simplest radical form.
ANSWER 4

13. GUIDED PRACTICE
for Example 1
Find the unknown side length of the right triangle. 2. 13
ANSWER

15. X = 3.2
Y = X + 5
Y =
Y = 8.2

16. 5.7²+ 8² = x²
= x²
96.49 = x²
X = 9.8
Y² = 32
Y = 5.7

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

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

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

20. 2²+ 2² = x²
4 + 4 = x²
x² = 8
X = 2.8

21. 30²+ 16² = x²
= x²
x² = 1156
X = 34

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

23. 14² + 48² = 50²
50² + 75² = 85²
15² + 36² = 39²
45² + 60² = 80² ?