1. Geometric Mean and Pythagorean Theorem
Lesson 8-1
Geometric Mean and Pythagorean Theorem
Lesson 7-1: Geometric Mean

2. Lesson 7-1: Geometric Mean
Sequences
Arithmetic Sequence:
Is a pattern of numbers where any term (number in the sequence) is determined by adding or subtracting the previous term by a constant called the common difference.
Example:
2, 5, 8, 11, 14, ____, ____, ____
Common difference = 3
Geometric Sequence:
Is a pattern of numbers where any term (number in the sequence) is determined by multiplying the previous term by a common factor.
Example:
2, 6, 18, 54, 162, _____, _____, ____
Common Factor = 3
Lesson 7-1: Geometric Mean

3. Lesson 7-1: Geometric Mean
Examples
1. Starting with the number 1 and using a factor of 4, create 5 terms of a geometric sequence.
1 , 4 , 16 , 64 , 256
2. Starting with the number 2 and using a factor of 5, create 5 terms of a geometric sequence.
2 , 10 , 50 , 250 , 1250
3. Starting with the number 5 and using a factor of 3, create 5 terms of a geometric sequence.
5 , 15 , 45 , 135 , 405
4. In the geometric sequence 2, ____, 72, 432, .Find the missing term. 12 12
5. In the geometric sequence 6, ____, 24,... Find the missing term.
Lesson 7-1: Geometric Mean

4. Lesson 7-1: Geometric Mean
A term between two terms of a geometric sequence is the geometric mean of the two terms.
Example:
In the geometric sequence 4, 20, 100, ….(with a factor of 5), 20 is the geometric mean of 4 and 100.
Try It:
Find the geometric mean of 3 and 300.
3 , ___ , 300 30
Lesson 7-1: Geometric Mean

5. Lesson 7-1: Geometric Mean
Geometric Mean : Fact
Consecutive terms of a geometric sequence are proportional.
Example:
Consider the geometric sequence with a common factor 10.
4 , 40 , 400
(4)(400) = (40)(40)
cross-products are equal
1600 = 1600
Lesson 7-1: Geometric Mean

6. Lesson 7-1: Geometric Mean
Therefore ………..
To find the geometric mean between 7 and
label the missing term x
7 , ___ , 28
write a proportion
cross multiply
solve
Lesson 7-1: Geometric Mean

7. Lesson 7-1: Geometric Mean
The geometric mean between two numbers a and b is the positive number x where Therefore x =
Try It:
Find the geometric mean of . . .
Answer = 20
and 40
2. 1 and 36
Answer = 6
Lesson 7-1: Geometric Mean

8. How does this relate to geometry?
Lesson 7-1: Geometric Mean

9. Lesson 7-1: Geometric Mean
The " W " Pattern
Lesson 7-1: Geometric Mean

10. Re-label the Sides (as lengths)
Lesson 7-1: Geometric Mean

11. Lesson 7-1: Geometric Mean
What is the proportion that uses f?
f is the geometric mean of d and e.
Lesson 7-1: Geometric Mean

12. Lesson 7-1: Geometric Mean
What is the proportion that uses b?
b is the geometric mean of e and c.
Lesson 7-1: Geometric Mean

13. Lesson 7-1: Geometric Mean
What is the proportion that uses a?
a is the geometric mean of d and c.
Lesson 7-1: Geometric Mean

14. Lesson 7-1: Geometric Mean
Put them all together
Lesson 7-1: Geometric Mean

15. Lesson 7-1: Geometric Mean
The “W” Pattern W
Lesson 7-1: Geometric Mean

16. Lesson 7-1: Geometric Mean
Try it !
Given: d = 4 and e = 10
Find:
a = ___
b = ___
c = ___
f = ___
Lesson 7-1: Geometric Mean

17. Lesson 7-1: Geometric Mean
Solution:
Proportions
Answers
Lesson 7-1: Geometric Mean

18. Lesson 7-2: The Pythagorean Theorem
Given the lengths of three sides, how do you know if you have a right triangle?
Given A = 6, B=8, and C=10, describe the triangle.
A2 + B2 = C2
= 102
= 100
This is true, so you have a right triangle. C A B
Lesson 7-2: The Pythagorean Theorem

19. If A2 + B2 > C2, you have an acute triangle.
Given A = 4, B = 5, and C =6, describe the triangle.
A2 + B2 = C2
= 62
= 36
41 > 36, so we have an acute triangle. A B C
Lesson 7-2: The Pythagorean Theorem

20. If A2 + B2 < C2, you have an obtuse triangle.
Given A = 4, B = 6, and C =8, describe the triangle.
A2 + B2 = C2
= 82
= 64
52 < 64, so we have an obtuse triangle. A B C
Lesson 7-2: The Pythagorean Theorem