1. 9.4 Special Right Triangles
CCSS: G.SRT.6

2. CCSS: G.SRT.6:
UNDERSTNAD that by similarity, side ratios in right triangles are properties of the angles in the triangle, LEADING to definitions of trigonometric ratios for acute angles.

3. Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.  
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

4. Objectives Find the side lengths of special right triangles.
Use special right triangles to solve real-life problems, such as finding the side lengths of the triangles.

5. How can we find the lengths of a special right triangle?
E.Q:
How can we find the lengths of a special right triangle?

6. 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.

7. Theorem 9.8: 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

8. Theorem 9.8: 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

9. 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

10. 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

11. 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

12. 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

13. 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°

14. 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

15. 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

16. 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?

17. 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.

18. 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.

19. Ex. 5: Finding the area of a sign
Road sign. The road sign is shaped like an equilateral triangle. Estimate the area of the sign by finding the area of the equilateral triangle.
18 in. h
36 in.

20. Ex. 5: Solution
18 in.
First, find the height h of the triangle by dividing it into two 30°-60°-90° triangles. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 18 inches.
h = √3 ∙ 18 = 18√3
30°-60°-90° Triangle Theorem h
36 in.
Use h = 18√3 to find the area of the equilateral triangle.

21. Ex. 5: Solution Area = ½ bh = ½ (36)(18√3) ≈ 561.18
18 in.
Area = ½ bh
= ½ (36)(18√3) ≈ h
36 in.
The area of the sign is a bout 561 square inches.