9 2 Special Right Triangles Geometry ObjectivesAssignment Find

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°-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°-90° Triangle Theorem 45°

Theorem 9. 8: 30°-60°-90° Triangle Theorem 60° 30°

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

Ex. 2: Finding a leg in a 45°-90° Triangle 5 x x

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

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 = 2 h 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°-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 • 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 18 in. h 36 in. Use h = 18√ 3 to find the area of the equilateral triangle.

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