8-1 Geometric Mean The student will be able to: 1.Find the geometric mean between two numbers. 2.Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse.

Geometric Mean The geometric mean of two positive numbers a and b is the positive square root of their product (x). The geometric mean is created when a proportion is set up, cross- multiplied and solved for x. x(x) = a(b) x 2 = ab The geometric mean of a = 9 and b = 4 is found by: x = 6 | |9

Example 1: Find the geometric mean between 5 and 45. x = 5(3) x = 15 | |

Geometric Means in Right Triangles In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two additional right triangles. These three right triangles are all similar. A D C C D B 1 st – Draw the two smaller triangles to look like the original. (Hint: match the right angles then match the shorter side) 2 nd – Write similarity statements for the three triangles. ΔACB~ΔADC~ΔCDB

Example 2: Write a similarity statement identifying the three similar right triangles in the figure. M K P M P L 1 st – Draw the two smaller triangles to look like the original. (Hint: match the right angles then match the shorter side) 2 nd – Write similarity statements for the three triangles. ΔKML~ΔKPM~ΔMPL

Example 3: Write a similarity statement identifying the three similar right triangles in the figure. T Q S S R T 1 st – Draw the two smaller triangles to look like the original. (Hint: match the right angles then match the shorter side) 2 nd – Write similarity statements for the three triangles. ΔQSR~ΔQTS~ΔSTR

You Try It: 1. Find the geometric mean between 2 and Write a similarity statement identifying the three similar triangles in the figure. 10 ΔEFG~ΔEHF~ΔFHG | | x = 5(2) x = 10 H E F F G H

Since we know that an altitude drawn to the hypotenuse of a right triangle forms 3 similar triangles, you can write proportions comparing the side lengths of these triangles. The Geometric Mean (Altitude) Theorem: The length of the altitude is the geometric mean between the two lengths of the hypotenuse or The Geometric Mean (leg) Theorem: The length of the leg is the geometric mean between the hypotenuse and the segment adjacent to that leg or or

Example 4: Find x, y, and z. or or 16.2 or 28.7 |8 | | |8 | | | |

Example 5: Find x, y, and z. 16 = x x + 9 y = 20 z = 15 | |16 |8 2 2 | | | | x

You Try It: 1. Find c, d, and e. 30 |24 3 |30 | | |8 2 |24 3 | | |30 |