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Special Right Triangles

Special Right Triangles. Section 9.4. 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. . Theorem 9.8: 45 °-45°-90° Triangle Theorem. In a 45 °-45°-90° triangle,

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Special Right Triangles

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  1. Special Right Triangles Section 9.4

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

  4. Theorem 9.9: 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. x Hypotenuse = 2 ∙ shorter leg Longer leg = ∙ shorter leg

  5. Example 1: Find x 3 3 45 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.

  6. Example 2: • Find the value of x. 5 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. x x

  7. Example 3: • Find the values of s and t. 60 Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. 30

  8. Longer leg = times shorter leg • 5 = times s • =s • 5 = s • Hypotenuse = 2 ∙ shorter leg

  9. Examples: Find x and y • 1. • 2.

  10. 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?

  11. 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.

  12. 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 • 40 • (56.6 ≈ h)

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