8.2 Special Right Triangles Geometry

Objectives/DFA/HW Objectives SWBAT use properties of 45o-45o-90o & 30o-60o-90o triangles. Why? Ex. – To find the distance from home plate to 2nd on a baseball diamond. DFA – p.504 #18 HW – pp.503-505 (2-32 even, 34-37 all)

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.

Theorem 8.5: 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. 45° √2x 45° Hypotenuse = √2 ∙ leg

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle Find the value of x By the Triangle Sum Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. 3 3 45° x

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x 45°-45°-90° Triangle Theorem Substitute values Simplify Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2

Ex. 2: Finding a leg in a 45°-45°-90° Triangle Find the value of x. Because the triangle is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°- 90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg. 5 x x

Ex. 2: Finding a leg in a 45°-45°-90° Triangle Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem Substitute values 5 √2x = Divide each side by √2 √2 √2 5 = x Simplify √2 Multiply numerator and denominator by √2 √2 5 = x √2 √2 5√2 Simplify = x 2

Theorem 8.6: 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. 60° 30° √3x Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

Ex. 3: Finding side lengths in a 30°-60°-90° Triangle Find the values of s and t. Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. 60° 30°

Ex. 3: Side lengths in a 30°-60°-90° Triangle Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem Substitute values 5 √3s = Divide each side by √3 √3 √3 5 = s Simplify √3 Multiply numerator and denominator by √3 √3 5 = s √3 √3 5√3 Simplify = s 3

Statement: Reasons: Substitute values Simplify The length t of the hypotenuse is twice the length s of the shorter leg. 60° 30° Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem 5√3 t 2 ∙ Substitute values = 3 10√3 Simplify t = 3

Using Special Right Triangles in Real Life Example 4: Finding the height of a ramp. Tipping platform. 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 30°-60°-90° Triangle Theorem 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. 80 = √2 ∙ h 45°-45°-90° Triangle Theorem 80 = h Divide each side by √2 √2 56.6 ≈ h Use a calculator to approximate When the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.

Ex. 5: Finding the distance from home plate to 2nd base on a baseball field A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between home plate and second base?