GOAL 1 PROPORTIONS IN RIGHT TRIANGLES

EXAMPLE 1

9.1 Similar Right Triangles

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.

Extra Example 1A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section.a. Identify the similar triangles in the diagram.b. Find the height h of the roof.

7.8 m 12.3 m

14.6 mA

B

CD

h

Checkpoint

6.8 in.

12.7 in.

14.4 in.

S

RU

T

h

The diagram shows the approximate dimensions of a right triangle.a. Identify the similar triangles in the diagram.b. Find the height h of the triangle.

GOAL 2 USING A GEOMETRIC MEAN TO SOLVE PROBLEMS

EXAMPLE 2

9.1 Similar Right Triangles

Study the Geometric Mean Theorems on page 529 before going on!

Extra Example 2

Find the value of each variable.

a. b.

6 10

x

5

8y

CheckpointFind the value of each variable.

a. b.

1424

x

5

18

y

GOAL 2 USING THE PYTHAGOREAN THEOREM

EXAMPLE 1

9.2 The Pythagorean Theorem

If ΔABC is a right triangle, then c2 = a2 + b2.

a

b

c

A

B

C

Extra Example 1Find the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple.

7

24

x

CheckpointFind the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple.

x

2

3

2

Extra Example 2

12 x

6 5

Find the length of the leg of the right triangle.

Checkpoint

Find the length of the leg of the right triangle.

921

x

Extra Example 3

h

8 m 8 m

10 m

Find the area of the triangle tothe nearest tenth of a meter.

Extra Example 4The two antennas shown in the diagram are supported by cables 100 feet in length. If the cables are attached to the antennas 50 feet from the ground, how far apart are the antennas?

Checkpoint

Find the missing side of the triangle. Then find the area to the nearest tenth of a meter.

12 m 12 m

10.8 m