1. Objective: To use the properties of 45°-45°-90° triangles.
Chapter 7 Lesson 3
Objective: To use the properties of °-45°-90° triangles.

2. hypotenuse leg leg x√2 x x Theorem 7-8: 45°-45°-90° Triangle Theorem
In a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is √2 times the length of a leg.
Hypotenuse = √2•leg
hypotenuse
45°
leg
45°
leg
x√2
45° x
45° x

3. Example 1: Finding the Length of the Hypotenuse
Find the value of each variable. a. b.
45° 9
45°
2√2 x h
h=√2•9
h=9√2
x=√2•2√2
x=4

4. Example 2: Finding the Length of the Hypotenuse
Find the length of the hypotenuse of a 45°-45°-90° triangle with legs of length 5√3.
5√3
5√3
x=√2•5√3
x=5√6
45° x

5. Example 3: Finding the Length of the Hypotenuse
Find the length of the hypotenuse of a 45°-45°-90° triangle with legs of length 5√6.
5√6
5√6
x=√2•5√6
x=5√12
x=5√(4•3)
x=10√3
45° x

6. Example 4: Finding the Length of a Leg
Find the value of x.
6=√2•x
45° 6 x

7. Example 5: Finding the Length of a Leg
Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 10.
10=√2•x
45° 10 x

8. Example 6: Finding the Length of a Leg
Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.
22=√2•x
45° 22 x

9. Theorem 7-9: 30°-60°-90° Triangle Theorem
In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is √3 times the length of the shorter leg.
hypotenuse = 2 • shorter leg
longer leg = √3 • shorter leg
30°
60°
30°
60°
long leg
hypotenuse 2x
x√3
short leg x

10. Example 7: Finding the Lengths of the Legs
Find the value of each variable.
60°
30° x y 8
Shorter Leg
hypotenuse = 2 • shorter leg
8 = 2x
x = 4
Longer Leg
longer leg = √3 • shorter leg
y = x√3
y = 4√3

11. Example 8: Finding the Lengths of the Legs
Find the lengths of a 30°-60°-90° triangle with hypotenuse of length 12.
60°
30° y x 12
Shorter Leg
hypotenuse = 2 • shorter leg
12 = 2x
x = 6
Longer Leg
longer leg = √3 • shorter leg
y = x√3
y = 6√3

12. Example 9: Finding the Lengths of the Legs
Find the lengths of a 30°-60°-90° triangle with hypotenuse of length 4√3.
Shorter Leg
hypotenuse = 2 • shorter leg
4√3 = 2x
x = 2√3
60°
30° x y
4√3
Longer Leg
longer leg = √3 • shorter leg
y = x√3
y = 2√3•√3
Y=6

13. Example 10: Using the Length of a Leg
Find the value of each variable.
Shorter Leg
long leg = √3 • short leg
Hypotenuse
Hyp. = 2 • shorter leg 5
30°
60° x y

14. Example 11: Using the Length of a Leg
The shorter leg of a 30°-60°-90° has length √6. What are the lengths of the other sides? Leave your answers in simplest radical form.
Longer Leg
longer leg = √3 • shorter leg
Hypotenuse
hyp. = 2 • shorter leg x
30°
60° √6 y

15. Example 12: Using the Length of a Leg
The longer leg of a 30°-60°-90° has length 18. Find the length of the shorter leg and the hypotenuse.
Shorter Leg
long leg = √3 • short leg
Hypotenuse
hyp. = 2 • shorter leg 18
30°
60° x y

16. Assignment
Pg. 369
#1-29; 34-39