1. Section 8.1 – 8.2 Geometric Mean Pythagorean Theorem
Geometry
Section 8.1 – 8.2
Geometric Mean
Pythagorean Theorem

2. Today’s Agenda Geometric Mean in Right Triangles
Pythagorean Theorem in Right Triangles

3. Geometric Mean A Geometric Mean is a kind of average.
To find the Geometric Mean between two numbers, multiply them together and take the square root.
Example: Find the Geometric Mean of 5 and 20.

4. Geometric Mean as a Proportion
In a proportion if the means are equal, then that value is the geometric mean of the extremes:
x represents the Geometric Mean between a and b.
Example:

5. Geometric Mean This concept can used in Geometry.
One particular use is when dealing with right triangles.

6. Geometric Mean
We start with a Right Triangle

7. Geometric Mean
Let’s draw its altitude.

8. Geometric Mean We’ve now formed 2 more triangles – 3 in all!
What do these 3 triangles have in common?

9. Geometric Mean
Let’s consider the original diagram C A B D

10. Geometric Mean We’ll put the others up for reference A C A A B D C D C

11. Geometric Mean Let’s label the sides A C A a b h d e A B a c D c a d C

12. Geometric Mean We can use similarity properties to set up proportions:b h d e A B a c D c a d C b h D C C B D B h b e

13. Geometric Mean
To conclude, a, b, and h, can all be written as the Geometric Mean of two segments. a b h d e c

14. Geometric Mean Putting it in words:
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments.
The length of this altitude is the geometric mean between the lengths of these segments.
The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

15. Example 5 20 x y z

16. Pythagorean Theorem: SECTION 8.2

17. Example 1: If a = 9, b = 12, solve for c. YOU TRY IT!

18. Example #2: If a = 12, c = 20, solve for b. YOU TRY IT!

19. Example 3: If a = 6 and b = 15, solve for c
Example 3: If a = 6 and b = 15, solve for c. (Answer in simplest radical form) YOU TRY IT:

20. Determine if a set of numbers can be the measures of sides of a triangle: USING the TRIANGLE INEQUALITY THEOREM: Example: 7, 15, 21 You try it!

21. Classify a triangle as Acute, Obtuse, or Right: