1. Geometric Mean and the Pythagorean Theorem
6.7
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 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. Right Triangle Altitude Similar Triangles Theorem
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. Right Triangle Altitude Theorem 1
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. Right Triangle Altitude Theorem 2
If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and segment of the hypotenuse adjacent to that leg.

10. Example 4
Find a and b in ∆TGR.

11. 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.

12. Homework
6.7 RSG