1. 8.2 Special Right Triangles
Geometry

2. Objectives/DFA/HW Objectives
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 (2-32 even, all)

3. Side lengths of Special Right Triangles
Right 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.

4. Theorem 8.5: 45°-45°-90° Triangle Theorem
In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.
45°
√2x
45°
Hypotenuse = √2 ∙ leg

5. Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle
Find the value of x
By 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
45° x

6. Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle3 3
45° x
45°-45°-90° Triangle Theorem
Substitute values
Simplify
Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2

7. 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

8. Ex. 2: Finding a leg in a 45°-45°-90° Triangle
Statement:
Hypotenuse = √2 ∙ leg
5 = √2 ∙ x
Reasons:
45°-45°-90° Triangle Theorem
Substitute values 5
√2x =
Divide each side by √2 √2 √2 5 = x
Simplify √2
Multiply numerator and denominator by √2 √2 5 = x √2 √2
5√2
Simplify = x 2

9. Theorem 8.6: 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.
60°
30°
√3x
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg

10. 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.
60°
30°

11. 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
Substitute values 5
√3s =
Divide each side by √3 √3 √3 5 = s
Simplify √3
Multiply numerator and denominator by √3 √3 5 = s √3 √3
5√3
Simplify = s 3

12. Statement: Reasons: Substitute values Simplify
The length t of the hypotenuse is twice the length s of the shorter leg.
60°
30°
Statement:
Hypotenuse = 2 ∙ shorter leg
Reasons:
30°-60°-90° Triangle Theorem
5√3 t
2 ∙
Substitute values = 3
10√3
Simplify t = 3

13. 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?

14. 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 Theorem
40 = h Divide each side by 2.
When the angle of elevation is 30°, the ramp height is about 40 feet.

15. 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 = h
Divide each side by √2 √2
56.6 ≈ h
Use a calculator to approximate
When the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.

16. 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?