Similar Right TrianglesChapter 8

Activity: Investigating similar right triangles. Do in pairs or threes1. 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.

Theorem 8.1 (page 518) 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

Proportions in right triangles In chapter 7, 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

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 8.1, you know that ∆CBD~∆ACD~∆ABC.

CD BA D

C

A

C

BD

A

C

BD

They ALL look the same!

Similar = same shape, different size

Geometric Mean Theorems Theorem 8.2: 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 8.3: 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.

Geometric Mean Theorems

A

C

BD

BD

CD= CD

ADGM

If you want to find the Altitude… use Geometric Mean

CD is the Geometric Mean

of AD

and BD

Example: Use Geometric Mean to find the Altitude of the Triangle

6

x= x

3

36

x

18 = x2

√18 = x

√9 ∙ √2 = x

3 √2 = x

Geometric Mean Theorems

AB

CB= CB

DB

A

C

BD

GM

BA

If you want to find the side on the right side of the triangle… use Geometric Mean

CB is the Geometric Mean

of DB

and AB

Example: Find y (the right leg of the triangle) using Geometric Mean

5 + 2

y= y

2

y 5

2

14 = y2

7

y= y

2

√14 = y

y

5 2

Y is the leg on the right side

Rotate the triangle to help visualize what part you need

Geometric Mean Theorems

AB

AC= AC

AD

A

C

BD

GM

BA

If you want to find the side on the LEFT side of the triangle… use Geometric Mean

AC is the Geometric Mean

of AD

and AB

Example: Find y (the left leg of the triangle) using Geometric Mean

35

y= y

7

245 = y2

√245 = y Y is the leg on the Left side

y

7

35

7√5 = y

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.