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 x√2 45 °

WALLPAPER TILING The wallpaper in the figure can be divided into four equal square quadrants so that each square contains 8 triangles. What is the area of one of the squares if the hypotenuse of each 45°- 45°- 90° triangle measures millimeters?

The length of the hypotenuse of one 45°- 45°- 90° triangle is millimeters. The length of the hypotenuse is times as long as a leg. So, the length of each leg is 7 millimeters. The area of one of these triangles is or 24.5 millimeters. Answer: Since there are 8 of these triangles in one square quadrant, the area of one of these squares is 8(24.5) or 196 mm 2.

WALLPAPER TILING If each 45°- 45°- 90° triangle in the figure has a hypotenuse of millimeters, what is the perimeter of the entire square? Answer: 80 mm

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 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 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

Find QR.

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

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 it lies on a vertical gridline. 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