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9.4 Special Right Triangles CCSS: G.SRT.6 CCSS: G.SRT.6: • UNDERSTNAD that by similarity, side ratios in right triangles are properties of the angles in the triangle, LEADING to definitions of trigonometric ratios for acute angles. Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated reasoning. Objectives • 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. E.Q: • How can we find the lengths of a special right triangle? 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 9.8: 45°-45°-90° Triangle Theorem • In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. 45° √2x x 45° x Hypotenuse = √2 ∙ leg Theorem 9.8: 30°-60°-90° Triangle Theorem • In a 30°-60°-90° triangle, the 60° hypotenuse is 2x twice as long as x the shorter leg, and the longer leg 30° is √3 times as long √3x as the shorter leg. 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 • 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 Hypotenuse = √2 ∙ leg 45°-45°-90° Triangle Theorem x = √2 ∙ 3 Substitute values x = 3√2 Simplify 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° 5 Triangle x Statement: Reasons: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x 5 √2 5 √2 √2 √2 x 5 √2 5√2 2 = √2x √2 = x 45°-45°-90° Triangle Theorem Substitute values Divide each side by √2 Simplify = x Multiply numerator and denominator by √2 = x Simplify 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° t s 30° 5 Ex. 3: Side lengths in a 30°-60°-90° Triangle t 60° s 30° 5 Statement: Reasons: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s 5 √3 5 √3 √3 √3 5 √3 5√3 3 = √3s √3 = s 30°-60°-90° Triangle Theorem Substitute values Divide each side by √3 Simplify = s Multiply numerator and denominator by √3 = s Simplify The length t of the hypotenuse is twice the length s of the shorter leg. 60° t s 30° 5 Statement: Reasons: Hypotenuse = 2 ∙ shorter leg 5√3 = 2∙ 3 t t = 10√3 3 30°-60°-90° Triangle Theorem Substitute values Simplify 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 80 √2 = h 45°-45°-90° Triangle Theorem Divide each side by √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 area of a sign • Road sign. The road sign is shaped like an equilateral triangle. Estimate the area of the sign by finding the area of the equilateral triangle. 18 in. h 36 in. Ex. 5: Solution • First, find the height h of the triangle by dividing it into two 30°-60°-90° triangles. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 18 inches. h = √3 ∙ 18 = 18√3 30°-60°-90° Triangle Theorem 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 a bout 561 square inches.