2. 8.2 Special Right Triangles
Geometry

3. Objectives/Assignment
Find the side lengths of special right triangles.
Use special right triangles to solve real-life problems
Quiz Next Class Period over

4. Side lengths of Special Right Triangles
Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90° are called special right triangles.

5. Theorem 8.6: 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

6. 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.
60°
30°
√3x
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg

7. 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
45° x
Hypotenuse = √2 ∙ leg

8. 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

9. 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

10. 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

11. 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°

12. 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

13. 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

14. X = 7 Y = 7√3 Hypotenuse = 2 ∙ shorter leg
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

15. Y= 5 x = 10 Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg
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

16. X = Y = 8√3 Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg
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

17. Hypotenuse = √2 ∙ leg X =6 √2 Y = √2*6 √2 Y = 6√4 Y = 6*2 Y = 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 =6 √2
Y = √2*6 √2
Y = 6√4
Y = 6*2
Y = 12

18. X =4 √3 Y = 2*4 =8 Hypotenuse = 2 ∙ shorter leg
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 =4 √3
Y = 2*4 =8

19. Hypotenuse = √2 ∙ leg 8 =√2* Leg Leg = 5.6 X =5.6 √3 = 9.8
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
8 =√2* Leg
Leg = 5.6
X =5.6 √3 = 9.8
Y = 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

20. 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
Diagonal = 10√2
Diagonal = 14.1 in