1. Concept

2. Find the Hypotenuse Length in a 45°-45°-90° Triangle
A. Find x.
The given angles of this triangle are 45° and 90°. This makes the third angle 45°, since 180 – 45 – 90 = 45. Thus, the triangle is a 45°-45°-90° triangle.
Example 1

3. 45°-45°-90° Triangle Theorem
Find the Hypotenuse Length in a 45°-45°-90° Triangle
45°-45°-90° Triangle Theorem
Substitution
Example 1

4. Find the Hypotenuse Length in a 45°-45°-90° Triangle
B. Find x.
The legs of this right triangle have the same measure, x, so it is a 45°-45°-90° triangle. Use the 45°-45°-90° Triangle Theorem.
Example 1

5. 45°-45°-90° Triangle Theorem
Find the Hypotenuse Length in a 45°-45°-90° Triangle
45°-45°-90° Triangle Theorem
Substitution
x = 12
Answer: x = 12
Example 1

6. A. Find x.
A. 3.5
B. 7 C. D. A B C D
Example 1

7. B. Find x. A. B.
C. 16
D. 32 A B C D
Example 1

8. 45°-45°-90° Triangle Theorem
Find the Leg Lengths in a 45°-45°-90° Triangle
Find a.
The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle.
45°-45°-90° Triangle Theorem
Substitution
Example 2

9. Rationalize the denominator.
Find the Leg Lengths in a 45°-45°-90° Triangle
Divide each side by
Rationalize the denominator.
Multiply.
Divide.
Example 2

10. Find b. A.
B. 3 C. D. A B C D
Example 2

11. Concept

12. Find Lengths in a 30°-60°-90° Triangle
Find x and y.
The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle.
Example 3

13. Find the length of the longer side.
Find Lengths in a 30°-60°-90° Triangle
Find the length of the longer side.
30°-60°-90° Triangle Theorem
Substitution
Simplify.
Example 3

14. Find the length of hypotenuse.
Find Lengths in a 30°-60°-90° Triangle
Find the length of hypotenuse.
30°-60°-90° Triangle Theorem
Substitution
Simplify.
Answer: x = 4,
Example 3

15. Find BC.
A. 4 in.
B. 8 in. C.
D. 12 in. A B C D
Example 3

16. Use Properties of Special Right Triangles
QUILTING A quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle?
Example 4

17. Use Properties of Special Right Triangles
Understand You know that the length of the side of the square equals 3 inches. You need to find the length of the side and hypotenuse of one isosceles right triangle.
Plan Find the length of one side of the isosceles right triangle, and use the 45°-45°-90° Triangle Theorem to find the hypotenuse.
Example 4

18. So the side length is 1.5 inches.
Use Properties of Special Right Triangles
Solve Divide the length of the side of the square by 2 to find the length of the side of one triangle. 3 ÷ 2 = 1.5
So the side length is 1.5 inches.
45°-45°-90° Triangle Theorem
Substitution
Example 4

19. Answer: The side length is 1.5 inches and the hypotenuse is
Use Properties of Special Right Triangles
Answer: The side length is 1.5 inches and the hypotenuse is
Check Use the Pythagorean Theorem to check the dimensions of the triangle. ?
= 4.5 ?
4.5 = 4.5 
Example 4

20. BOOKENDS Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends. A.
B. 10
C. 5 D. A B C D
Example 4