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9.4 Special Right Triangles

9.4 Special Right Triangles. Geometry Spring 2011. Objectives/Assignment. Find the side lengths of special right triangles. Use special right triangles to solve real-life problems. What are Special Right Triangles?. Right triangles whose angle measures are 45 °-45°-90° or 30°-60°-90°.

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9.4 Special Right Triangles

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  1. 9.4 Special Right Triangles Geometry Spring 2011

  2. Objectives/Assignment • Find the side lengths of special right triangles. • Use special right triangles to solve real-life problems.

  3. What are Special Right Triangles? • Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90°.

  4. 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

  5. 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

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

  7. Find the value of x. Ex. 2: Finding a leg in a 45°-45°-90° Triangle 5 x x

  8. Find the values of s and t. Ex. 3: Finding side lengths in a 30°-60°-90° Triangle 60° 30°

  9. 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?

  10. 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.

  11. 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.

  12. 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.

  13. 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.

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

  15. Reminders: Chapter 9 Quiz Tomorrow!! Sections 9.1-9.3

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