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

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

3. EXAMPLE 4
Find a height using indirect measurement
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.
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. Find a height using indirect measurement
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.
8.5 w = 5
Write a proportion. w
Solve for w.
So, the height of the wall is 5 + w = 19.5 feet.

5. GUIDED PRACTICE
for Examples 3 and 4
In Example 3, which theorem did you use to solve for y? 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. GUIDED PRACTICE
for Examples 3 and 4
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 x ft away from the wall because your new wall height w is 14.0 ft.
14.0 x
5.5 =
x2 = 77
x = 8.77 ft