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9.4 Special Right Triangles

9.4 Special Right Triangles. Geometry NCSCOS: 1.02; 2.03. 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?.

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9.4 Special Right Triangles

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  1. 9.4 Special Right Triangles Geometry NCSCOS: 1.02; 2.03

  2. 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.

  3. E.Q: • How can we find the lengths of a special right triangle?

  4. 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.

  5. In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. Theorem 9.8: 45°-45°-90° Triangle Theorem 45° √2x 45° Hypotenuse = √2 ∙ leg

  6. 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. Theorem 9.8: 30°-60°-90° Triangle Theorem 60° 30° √3x Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

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

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

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

  10. Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem Ex. 2: Finding a leg in a 45°-45°-90° Triangle 5 x x 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

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

  12. Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem Ex. 3: Side lengths in a 30°-60°-90° Triangle 60° 30° 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

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

  14. 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?

  15. 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.

  16. 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.

  17. 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. Ex. 5: Finding the area of a sign 18 in. h 36 in.

  18. 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 Use h = 18√3 to find the area of the equilateral triangle. Ex. 5: Solution 18 in. h 36 in.

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

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