9.2 Special Right Triangles Geometry

Objectives/Assignment Find the side lengths of special right triangles. Use special right triangles to solve real-life problems, such as finding the side lengths of the triangles.

Side lengths of Special Right Triangles Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow in this lesson.

Theorem 9.8: 45°-45°-90° Triangle Theorem

Theorem 9.8: 30°-60°-90° Triangle Theorem

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x

Ex. 2: Finding a leg in a 45°-45°-90° Triangle

Ex. 3: Finding side lengths in a 30°-60°-90° Triangle

Using Special Right Triangles in Real Life Example 4: Finding the height of a ramp. A tipping platform is a ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?

Solution: When the angle of elevation is 30°, the height of the ramp is the length of the shorter leg of a 30°-60°-90° triangle. The length of the hypotenuse is 80 feet. 80 = 2h 40 = h Divide each side by 2. When the angle of elevation is 30°, the ramp height is about 40 feet.

Solution: When the angle of elevation is 45°, the height of the ramp is the length of a leg of a 45°-45°-90° triangle. The length of the hypotenuse is 80 feet. 56.6 ft ≈ h

Ex. 5: Finding the area of a sign The road sign is shaped like an equilateral (equiangular) triangle. Estimate the area of the sign 18 in. h

Ex. 5: Solution 18 in. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 18 inches. h = 18√3 h 36 in. Use h = 18√3 to find the area of the equilateral triangle.

Area = ½ bh = ½ (36)(18√3) ≈ 561.18 Ex. 5: Solution 18 in. h 36 in. The area of the sign is about 561 square inches.