8.1 Geometric Mean The geometric mean between two numbers is the positive square root of their product. Another way to look at it… The geometric mean is the positive number x where the proportion a:x = x:b is true. Cross Multiply

Example 1 Find the geometric mean between the pair of numbers. 4 and 9 Try this one. 8 and 10 Answer:

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 triangles formed are similar to the given triangle and to each other. A B D C

Theorem 8.2 The measure of an altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measure of the two segments of the hypotenuse. A B D C BD is the geometric mean of AD and CD.

Example 2 In ABC, CD = 3 and AD = 14. Find BD. Now try this one: In ABC, CD = 0.8 and AD = 2.2. Find BD. Answer: 1.3 B A D C

Theorem 8.3 If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the measure of the leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. A D C B

Example 3 A D C B Find x and y in ABC. y x 14 2