8-3 Special Right Triangles

You used properties of isosceles and equilateral triangles.

• Use the properties of 45°-45°-90° triangles.

• Use the properties of 30°-60°-90° triangles.

Making of an Isosceles Right Triangle #1

How can you make an isosceles right triangle?

Right Ratios

Use the Pythagorean Theorem to find the third side.

27

45°

45°

7

7

4

4

24

45°- 45°- 90° Right TriangleIn a 45°- 45°- 90° triangle, the

hypotenuse is √2 times as long as either leg. The ratios of the side lengths can be written l-l-l√2.

l

l

2l

p. 558

Find the length of the side

210x

24

10x

r

s

a

b

9

l

l

l

l

2

29

229

2292

29r = s = 4

Find x.

The given angles of this triangle are 45° and 90°. This makes the third angle 45°, since 180 – 45 – 90 = 45. Thus, the triangle is a 45°-45°-90° triangle.

Substitution

45°-45°-90° Triangle Theorem

Find x.

The legs of this right triangle have the same measure, x, so it is a 45°-45°-90° triangle. Use the 45°-45°-90° Triangle Theorem.

Substitution

45°-45°-90° Triangle Theorem

x = 12

Answer: x = 12

Find x.

A. 3.5

B. 7

C.

D.

Find x.

A.

B.

C. 16

D. 32

Find a.

The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle.

Substitution

45°-45°-90° Triangle Theorem

Multiply.

Divide.

Rationalize the denominator.

Divide each side by

Making of an Isosceles Right Triangle #2

How can you make an isosceles right triangle?

60° 60°

60°

Right Ratios

Use the Pythagorean Theorem to find the third side.

12 12

6 6

?

60° 60°

30°

36?

30°- 60° - 90° Right Triangle In a 30°- 60° - 90° triangle, the

hypotenuse is twice as long as the shorter leg (the leg opposite the 30° angle, and the longer leg (opposite the 60° angle) is √3 tunes as long as the shorter leg. The ratios of the side lengths can be written l - l√3 – 2l.

60°

30°

l

2l

3l

p. 560

Find the length of the side

353 l

34l 3103 l

30°

60°

30°

60°30°

60°

30°

60°

4

20

4.8

2l = 8

2l = 20

l = 10l = 10

l = 52l = 10

77.23

8.38.4

38.43

38.433

8.43

l

l

l

l

l

2.77

5.54

Find BC.

A. 4 in.

B. 8 in.

C.

D. 12 in.

BOOKENDS Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends.

A.

B. 10

C. 5

D.

What two type of right triangles occur often?30°-60°-90° and 45°-45°-90°.

How can you find the length of a side of a special right triangle knowing only one side?

60°

30°

l

2l

3l

l

l

2l

8-3 AssignmentWorksheet 5-3B

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