9.1 Similar Right Triangles

Proportions in right triangles In Lesson 8.4, you learned

that two triangles are similar if two of their corresponding angles are congruent. For example ∆PQR ~ ∆STU. Recall that the corresponding side lengths of similar triangles are in proportion.

P

R Q

S

U T

Activity: Investigating similar right triangles. Group of two or three.1. Cut an index card along one of

its diagonals. 2. On one of the right triangles,

draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles.

3. You should now have three right triangles. Compare the triangles. What special property do they share? Explain.

4. Tape your group’s triangles to a piece of paper and place in homework bin.

What did you discover? In the activity, you may have discovered the

next theorem.

Theorem 9.1 If the altitude is drawn

to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. A B

C

D

∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD

Ex. 1: Finding the Height of a Roof Roof Height. A roof has a

cross section that is a right angle. The diagram shows the approximate dimensions of this cross section.

A. Identify the similar triangles.

B. Find the height h of the roof.

Solution: You may find it helpful to

sketch the three similar triangles so that the corresponding angles and sides have the same orientation. Mark the congruent angles. Notice that some sides appear in more than one triangle. For instance XY is the hypotenuse in ∆XYW and the shorter leg in ∆XZY.

h3.1 m

Y

X W

h

5.5 m

Z

Y W

5.5 m

3.1 m

6.3 m

Z

X Y

∆XYW ~ ∆YZW ~ ∆XZY.

Solution for b. Use the fact that ∆XYW ~ ∆XZY to write a

proportion. YW

ZY= XY

XZ

h

5.5= 3.1

6.3

6.3h = 5.5(3.1)

h ≈ 2.7

The height of the roof is about 2.7 meters.

Corresponding side lengths are in proportion.

Substitute values.

Cross Product property

Solve for unknown h.

Using a geometric mean to solve problems In right ∆ABC, altitude

CD is drawn to the hypotenuse, forming two smaller right triangles that are similar to ∆ABC From Theorem 9.1, you know that ∆CBD ~ ∆ACD ~ ∆ABC.

A

C

BD

C D

B

A D

C

A C

B

Write this down!

A

C

BD

C D

B

A D

C

A C

B

Notice that CD is the longer leg of ∆CBD and the shorter leg of ∆ACD. When you write a proportion comparing the legs lengths of ∆CBD and ∆ACD, you can see that CD is the geometric mean of BD and AD.

BD

CD= CD

AD

Shorter leg of ∆CBD.

Shorter leg of ∆ACD

Longer leg of ∆CBD.

Longer leg of ∆ACD.

Copy this down!

A

C

BD

C D

B

A D

C

A C

B

Sides CB and AC also appear in more than one triangle. Their side lengths are also geometric means, as shown by the proportions below:

AB

CB= CB

DB

Hypotenuse of ∆ABC.

Hypotenuse of ∆CBD

Shorter leg of ∆ABC.

Shorter leg of ∆CBD.

Geometric Mean Theorems Theorem 9.2: In a right triangle, the

altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments

Theorem 9.3: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

A

C

BD

BD

CD= CD

AD

AB

CB= CB

DB

AB

AC= AC

AD

What does that mean?

6

x= x

3

5 + 2

y= y

2

36

x y 5

2

18 = x2

√18 = x

√9 ∙ √2 = x

3 √2 = x

14 = y2

7

y= y

2

√14 = y

Ex. 3: Using Indirect Measurement. MONORAIL TRACK. To

estimate the height of a monorail track, your friend holds a cardboard square at eye level. Your friend lines up the top edge of the square with the track and the bottom edge with the ground. You measure the distance from the ground to your friend’s eye and the distance from your friend to the track.

In the diagram, XY = h – 5.75 is the difference between the track height h and your friend’s eye level. Use Theorem 9.2 to write a proportion involving XY. Then you can solve for h.