1. 1. Are these triangles similar? If so, give the reason.
ANSWER
Yes; the AA Similarity Postulate

2. 2. Find x.
ANSWER 50

3. EXAMPLE 1
Identify similar triangles
Identify the similar triangles in the diagram.
SOLUTION
Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation.
TSU ~ RTU ~ RST

4. EXAMPLE 2
Find the length of the altitude to the hypotenuse
Swimming Pool
The diagram below shows a cross-section of a swimming pool. What is the maximum depth of the pool?

5. EXAMPLE 2
Find the length of the altitude to the hypotenuse
SOLUTION
STEP 1
Identify the similar triangles and sketch them.
RST ~ RTM ~ TSM

6. Find the length of the altitude to the hypotenuse
EXAMPLE 2
Find the length of the altitude to the hypotenuse
STEP 2
Find the value of h. Use the fact that RST ~ RTM to write a proportion. ST TM = SR TR
Corresponding side lengths of similar triangles are in proportion. 64 h =
165
152
Substitute.
165h = 64(152)
Cross Products Property h
Solve for h.
STEP 3
Read the diagram. You can see that the maximum depth of the pool is h + 48, which is about = 107 inches.
The maximum depth of the pool is about 107 inches.

7. GUIDED PRACTICE
for Examples 1 and 2
Identify the similar triangles. Then find the value of x. 1.
EGF ~ GHF ~ EHG ; 12 5
ANSWER 2.
LMJ ~ MKJ ~ LKM ; 60 13
ANSWER

8. 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.

9. 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.

10. 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.

11. 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.

12. GUIDED PRACTICE
for Examples 3 and 4
3. In Example 3, which theorem did you use to solve for y ? Explain.
Theroem 7.7; This was used to set the ratios of the hypotenuse of the large triangle to the shorter leg and the hypotenuse of the small triangle to the shorter leg equal to each other.
ANSWER

13. 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?
ANSWER
about 8.93 ft

14. Daily Homework Quiz 1.
Identify the three similar right triangles in the diagram.
ANSWER
XYZ ~ XWY ~ YWZ

15. Daily Homework Quiz 2.
Find the values of the variable.
ANSWER 3 5

16. Daily Homework Quiz 3.
Find the values of the variable. 2 15
ANSWER