7.3 Special Right Triangles

7.3 Special Right Triangles

Objectives • Use properties of 45° - 45° - 90° triangles • Use properties of 30° - 60° - 90° triangles

Side Lengths of Special Right ∆s • Right triangles whose angle measures are 45° - 45° - 90°or 30° - 60° - 90°are called special right triangles. The theorems that describe the relationships between the side lengths of each of these special right triangles are as follows:

45° - 45° - 90°∆ Theorem 7.6 In a 45°- 45°- 90° triangle, the length of the hypotenuse is √2 times the length of a leg. hypotenuse = √2 • leg x√2 45 ° 45 °

The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle. Example 2: Find a.

Divide each side by Answer: Example 2: Rationalize the denominator. Multiply. Divide.

Answer: Your Turn: Find b.

30° - 60° - 90°∆ Be sure you realize the shorter leg is opposite the 30°& the longer leg is opposite the 60°. Theorem 7.7 • In a 30°- 60°- 90° triangle, the length of the hypotenuse is twice as long as the shorter leg, and the length of the longer leg is √3 times as long as the shorter leg. 60 ° 30 ° x√3 Hypotenuse = 2 ∙ shorter legLonger leg = √3 ∙ shorter leg

Example 3: Find QR.

is the longer leg, is the shorter leg, and is the hypotenuse. Answer: Example 3: Multiply each side by 2.

Your Turn: Find BC. Answer: BC = 8 in.

7.3 Special Right Triangles