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Geometry 8.2 Special Right Triangles

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Objectives/DFA/HW Objectives SWBAT use properties of 45 o -45 o -90 o & 30 o -60 o -90 o triangles. Why? Ex. – To find the distance from home plate to 2 nd on a baseball diamond. DFA – p.504 #18 HW – pp (2-32 even, all)

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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.

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Theorem 8.5: 45 °-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. √2x 45 ° Hypotenuse = √2 ∙ leg

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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. 33 x 45 °

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Ex. 1: Finding the hypotenuse in a 45°-45°- 90° Triangle Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2 33 x 45 ° 45°-45°-90° Triangle Theorem Substitute values Simplify

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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

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Ex. 2: Finding a leg in a 45°-45°-90° Triangle Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem 5 x x 5 √2 √2x √2 = 5 x= 5 x= 5√2 2 x= Substitute values Divide each side by √2 Simplify Multiply numerator and denominator by √2 Simplify

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Theorem 8.6: 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. √3x 60 ° 30 ° Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

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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. 30 ° 60 °

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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 5 √3 √3s √3 = 5 s= 5 s= 5√3 3 s= Substitute values Divide each side by √3 Simplify Multiply numerator and denominator by √3 Simplify 30 ° 60 °

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The length t of the hypotenuse is twice the length s of the shorter leg. Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem t2 ∙ 5√3 3 = Substitute values Simplify 30 ° 60 ° t 10√3 3 =

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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?

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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.

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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 ∙ h45°-45°-90° Triangle Theorem 80 √2 =h Divide each side by √2 Use a calculator to approximate56.6 ≈ h When the angle of elevation is 45 °, the ramp height is about 56 feet 7 inches.

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Ex. 5: Finding the distance from home plate to 2 nd base on a baseball field A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between home plate and second base?

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