7.3 Special Right Triangles

 Use properties of 45° - 45° - 90° triangles  Use properties of 30° - 60° - 90° triangles

 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:

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 OR The legs and hypotenuse have a ratio of 1:1:√2. x√2 45 ° A BC

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

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

Find q. The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle. P Q R q ° PR is the hypotenuse so PR = q = 7 √2. Answer: q = 7 √2

Find b. Answer:

 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. Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg OR The legs and hypotenuse have a ratio of 1:√3:2. x√3 60 ° 30 ° Be sure you realize the shorter leg is opposite the 30°  & the longer leg is opposite the 60° . Theorem 7.7 A B C

Find QR.

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

Find PR.

is the longer leg, is the shorter leg, and is the hypotenuse. Answer: PR = 4 PR = √3 (4 √3 / 3) PR = 4√9 / 3 PR = 4 3 / 3 PR = 4

Find BC. Answer: BC = 8 in.

COORDINATE GEOMETRY is a 30°-60°-90° triangle with right angle X and as the longer leg. Graph points X(-2, 7) and Y(-7, 7), and locate point W in Quadrant III.

Graph X and Y. lies on a horizontal gridline of the coordinate plane. Since will be perpendicular to and in Quadrant III we know that it lies on a vertical gridline below X. First, find the length of

is the shorter leg. is the longer leg. So, Use XY to find WX. Point W has the same x-coordinate as X. W is located units below X. Answer: The coordinates of W are or about

COORDINATE GEOMETRY is at 30°-60°-90° triangle with right angle R and as the longer leg. Graph points T(3, 3) and R(3, 6) and locate point S in Quadrant III. Answer: The coordinates of S are or about

 Pre-AP Geometry: Pg. 360 #12 – 28, 36, 38, 44  Geometry: Pg. 360 #12 – 24, 36, and 38