EXAMPLE 3

1. Find the value of y. Write your answer in simplest radical form. Draw the three similar triangles. STEP 1 EXAMPLE 3 Use a geometric mean SOLUTION

2. Write a proportion. STEP 2 length of shorter leg of RPQ length of hyp. of RPQ = length of hyp. of RQS length of shorter leg of RQS = 9 y 3 y 27 = y 3 3 = y EXAMPLE 3 Use a geometric mean Substitute. 27 = y2 Cross Products Property Take the positive square root of each side. Simplify.

3. Rock Climbing Wall To find the cost of installing a rock wall in your school gymnasium, you need to find the height of the gym wall. EXAMPLE 4 Find a height using indirect measurement You use a cardboard square to line up the top and bottom of the gym wall. Your friend measures the vertical distance from the ground to your eye and the distance from you to the gym wall. Approximate the height of the gym wall.

4. 8.5 w = 5 8.5 w 14.5 So, the height of the wall is 5 + w 5+ 14.5 = 19.5 feet. EXAMPLE 4 Find a height using indirect measurement SOLUTION By Theorem 7.6, you know that 8.5 is the geometric mean of w and 5. Write a proportion. Solve for w.

5. for Examples 3 and 4 GUIDED PRACTICE 3.In Example 3, which theorem did you use to solve fory? Explain. SOLUTION In example 3, the theorem used was the geometric mean (leg) theorem This was used to set the ratios of the hypotenuse of the Lange triangle to the shorter leg and the hypotenuse of the small triangle to the shorter leg equal to each other

6. = 14.0 x x 5.5 for Examples 3 and 4 GUIDED PRACTICE 4. Mary is 5.5 feet tall. How far from the wall in Example 4 would she have to stand in order to measure its height? SOLUTION As per example 4, the height of wall is 19.5 feet Height of a Mary is 5.5 feet tall Let her stand xft away from the wall because your new wall height w is 14.0ft. x2 = 77 x = 8.77 ft