1. Geometric Mean and the Pythagorean Theorem
8-1
Geometric Mean and the Pythagorean Theorem
Objectives:
To find the geometric mean between two numbers.
To solve problems involving relationships between parts of a triangle and the altitude and its hypotenuse.
To use the Pythagorean Theorem and its converse.

2. Vocabulary
Geometric Mean
Pythagorean Theorem
Pythagorean Triple

3. Geometric Mean Find the geometric mean between 2 and 10.
Let x represent the geometric mean.

4. Example 1
Find the geometric mean between 12 and 20.

5. Example 2
Find the geometric mean between 6 and 15.

6. Theorem 8-1
If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two right triangles formed are similar to the given triangle and to each other.

7. Theorem 8-2
The measure of the altitude drawn from the vertex of a right angle to its hypotenuse is the geometric mean between measures of the two segments of the hypotenuse.

8. Example 3

9. Theorem 8-3

10. Example 4
Find a and b in ∆TGR.

11. Theorem 8-4 Pythagorean Theorem
In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

12. Theorem 8-5 Converse of the Pythagorean Theorem
If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

13. Pythagorean Triple
A Pythagorean Triple is a group of three whole numbers that satisfies the equation
a2 + b2 = c2, where c is the greatest measure.

14. Homework
8-1 Worksheet