Geometry

8.2 Special Right Triangles

Objectives/DFA/HWObjectives/DFA/HWObjectives SWBAT use properties of 45o-45o-90o &

30o-60o-90o triangles.Why? Ex. – To find the distance from

home plate to 2nd on a baseball diamond.

DFA – p.504 #18

HW – pp.503-505 (2-32 even, 34-37 all)

Side lengths of Special Right TrianglesRight triangles whose angle measures are

45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow.

Theorem 8.5: 45°-45°-90° Triangle TheoremIn a 45°-45°-90°

triangle, the hypotenuse is √2 times as long as each leg.

x

x√2x

45°

45°

Hypotenuse = √2 ∙ leg

Ex. 1: Finding the hypotenuse in a 45°-45°-90° TriangleFind the value of xBy the Triangle Sum

Theorem, the measure of the third angle is 45°. 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°

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.Because the triangle

is an isosceles right triangle, its base angles are congruent. 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

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

Statement:Hypotenuse = √2 ∙ leg

5 = √2 ∙ x

Reasons:45°-45°-90° Triangle Theorem

5

x x

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

Theorem 8.6: 30°-60°-90° Triangle TheoremIn 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. 3: Finding side lengths in a 30°-60°-90° TriangleFind 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=

Using Special Right Triangles in Real Life

Example 4: Finding the height of a ramp.Tipping platform. A tipping platform is a

ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?

Solution:When the angle of elevation is 30°, the

height of the ramp is the length of the shorter leg of a 30°-60°-90° triangle. The length of the hypotenuse is 80 feet.

80 = 2h 30°-60°-90° Triangle Theorem40 = h Divide each side by 2.

When the angle of elevation is 30°, the ramp height is about 40 feet.

Solution:When the angle of elevation is 45°, the

height of the ramp is the length of a leg of a 45°-45°-90° triangle. The length of the hypotenuse is 80 feet.

80 = √2 ∙ h 45°-45°-90° Triangle Theorem

80

√2= h Divide each side by √2

Use a calculator to approximate56.6 ≈ hWhen the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.

Ex. 5: Finding the distance from home plate to 2nd base on a baseball field

A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between home plate and second base?