8 1 Geometric Mean The student will be

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, crossmultiplied and solved for x. The geometric mean of a = 9 and b = 4 is found by: 2 |4 2 2 3 |9 3 x=6 3 x(x) = a(b) x 2 = ab

Example 1: Find the geometric mean between 5 and 45. x = 5(3) x = 15 5 3 5 5 |45 3 |9 3

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. 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 D D A C C B

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

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

You Try It: 1. Find the geometric mean between 2 and 50. 10 2 2 x = 5(2) 5 |50 x = 10 2 |10 5 5 2. Write a similarity statement identifying the three similar triangles in the figure. ΔEFG~ΔEHF~ΔFHG H E H F F G

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 or adjacent to that leg or

Example 4: Find x, y, and z. 33 2 or 14. 1 2 |8 2 |4 2 5 |25 5 5 2 or 16. 2 2 |8 2 |4 2 3 |33 11 5 |25 5 5 or 28. 7 3 |33 11

Example 5: Find x, y, and z. x+9 3 3 |9 3 x 16 = x y = 20 2 |16 2 2 |8 2 |4 2 2 5 |25 5 5 3 z = 15 3 |9 3 5 |25 5 5

You Try It: 1. Find c, d, and e. 3 |24 2 |8 2 |4 2 3 |6 2 2 3 |24 2 |8 2 |4 2 2 2 5 |30 3 3 |6 2 30 3 2 3 |6 2 5 |30 3 |6 2