8.2 Special Right Triangles

Geometry

Objectives/Assignment

• Find the side lengths of special right triangles.

• Use special right triangles to solve real-life problems

• Quiz Next Class Period over 8.1- 8.2

Side lengths of Special Right Triangles• Right triangles whose angle

measures are 45°-45°-90° or 30°-60°-90° are called special right triangles.

Theorem 8.6: 45°-45°-90° Triangle Theorem• In a 45°-45°-90° triangle, the

hypotenuse is √2 times as long as each leg.

x

x√2x

45°

45°

Hypotenuse = √2 ∙ leg

Theorem 8.7: 30°-60°-90° Triangle Theorem• In a 30°-60°-90° triangle, the

hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

√3x

60°

30°

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

2xx

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle

• Find the value of x• The triangle is a 45°-45°-90° right triangle, so the

length x of the hypotenuse is √2 times the length of a leg.

3 3

x

45°

Hypotenuse = √2 ∙ leg

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle

Hypotenuse = √2 ∙ leg

x = √2 ∙ 3

x = 3√2

3 3

x

45°

45°-45°-90° Triangle Theorem

Substitute values

Simplify

Ex. 2: Finding a leg in a 45°-45°-90° Triangle

• Find the value of x.• The triangle is a 45°-

45°-90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg.

5

x x

Hypotenuse = √2 ∙ leg

Ex. 2: Finding a leg in a 45°-45°-90° Triangle

5

x x

Statement:Hypotenuse = √2 ∙ leg

5 = √2 ∙ x

Reasons:45°-45°-90° Triangle Theorem

5

√2

√2x

√2=

5

√2x=

5

√2x=

√2

√2

5√2

2x=

Substitute values

Divide each side by √2

Simplify

Multiply numerator and denominator by √2

Simplify

Ex. 3: Finding side lengths in a 30°-60°-90° Triangle

• Find the values of s and t.• Because the triangle is a 30°-60°-

90° triangle, the longer leg is √3 times the length s of the shorter leg.

5

st

30°

60°

Ex. 3: Side lengths in a 30°-60°-90° Triangle

Statement:Longer leg = √3 ∙ shorter leg

5 = √3 ∙ s

Reasons:30°-60°-90° Triangle Theorem

5

√3

√3s

√3=

5

√3s=

5

√3s=

√3

√3

5√3

3s=

Substitute values

Divide each side by √3

Simplify

Multiply numerator and denominator by √3

Simplify

5

st

30°

60°

The length t of the hypotenuse is twice the length s of the shorter leg.

Statement:Hypotenuse = 2 ∙ shorter leg

Reasons:30°-60°-90° Triangle Theorem

t 2 ∙ 5√3

3= Substitute values

Simplify

5

st

30°

60°

t 10√3

3=

Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

X = 7 Y = 7√3

Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

Y= 5 x = 10

Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

X =

Y = 8√3

X =6 √2

Y = √2*6 √2 Y = 6√4Y = 6*2Y = 12

The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.

Hypotenuse = √2 ∙ leg

X =4 √3

Y = 2*4 =8

Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

8 =√2* LegLeg = 5.6

X =5.6 √3 = 9.8Y = 2*5.6 =11.4

Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.

Hypotenuse = √2 ∙ leg

Diagonal = 10√2 Diagonal = 14.1 in

The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.

Hypotenuse = √2 ∙ leg