8.2 Special Right Triangles Geometry

Objectives/Assignment Find the side lengths of special right triangles. Use special right triangles to solve real-life problems Quiz Next Class Period over 8.1- 8.2

Side lengths of Special Right Triangles Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90° are called special right triangles.

Theorem 8.6: 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

Theorem 8.7: 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. 1: Finding the hypotenuse in a 45°-45°-90° Triangle Find the value of x 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 Hypotenuse = √2 ∙ leg

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. 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 Hypotenuse = √2 ∙ leg

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

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

X = 7 Y = 7√3 Hypotenuse = 2 ∙ shorter leg Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg X = 7 Y = 7√3

Y= 5 x = 10 Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg Y= 5 x = 10

X = Y = 8√3 Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg X = Y = 8√3

Hypotenuse = √2 ∙ leg X =6 √2 Y = √2*6 √2 Y = 6√4 Y = 6*2 Y = 12 The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. Hypotenuse = √2 ∙ leg X =6 √2 Y = √2*6 √2 Y = 6√4 Y = 6*2 Y = 12

X =4 √3 Y = 2*4 =8 Hypotenuse = 2 ∙ shorter leg Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg X =4 √3 Y = 2*4 =8

Hypotenuse = √2 ∙ leg 8 =√2* Leg Leg = 5.6 X =5.6 √3 = 9.8 The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. Hypotenuse = √2 ∙ leg 8 =√2* Leg Leg = 5.6 X =5.6 √3 = 9.8 Y = 2*5.6 =11.4 Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

Hypotenuse = √2 ∙ leg Diagonal = 10√2 Diagonal = 14.1 in The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. Hypotenuse = √2 ∙ leg Diagonal = 10√2 Diagonal = 14.1 in