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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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