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Bell Ringer

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30-60-90 Triangles A Right Triangle with angle measures of 30, 60, and 90 are called 30-60-90 triangles.

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Example 1 Find Leg Length SOLUTION You can use the Pythagorean Theorem to find the value of b. (leg) 2 + (leg) 2 = (hypotenuse) 2 Write the Pythagorean Theorem. 1 2 + b 2 = 2 2 Substitute. 1 + b 2 = 4 Simplify. b 2 = 3 Subtract 1 from each side.Take the square root of each side. b =3 In the diagram, PQR is a 30° –60° –90° triangle with PQ = 2 and PR = 1. Find the value of b.

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Example 2 Find Hypotenuse Length SOLUTION The hypotenuse of a 30° –60° –90° triangle is twice as long as the shorter leg. hypotenuse = 2 · shorter leg 30° –60° –90° Triangle Theorem = 2 · 12 Substitute. = 24 Simplify. In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse. ANSWER The length of the hypotenuse is 24.

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Example 3 Find Longer Leg Length In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the longer leg. ANSWER The length of the longer leg is 5. 3 30° –60° –90° Triangle Theorem longer leg = shorter leg ·3 Substitute. = 5 · 3 SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times. 3

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Now You Try Find Lengths in a Triangle ANSWER 14 Find the value of x. Write your answer in radical form. 1. 2. 3. ANSWER 3 3 103

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Example 4 Find Shorter Leg Length In the 30° –60° –90° triangle at the right, the length of the longer leg is given. Find the length x of the shorter leg. Round your answer to the nearest tenth. 2.9 ≈ x Use a calculator. ANSWER The length of the shorter leg is about 2.9. Substitute. 5 = x ·3 30° –60° – 90° Triangle Theorem longer leg = shorter leg · 3 SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times 3. = x 5 3 Divide each side by. 3

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Example 5 Find Leg Lengths In the 30° –60° –90° triangle at the right, the length of the hypotenuse is given. Find the length x of the shorter leg and the length y of the longer leg. SOLUTION Use the 30° –60° –90° Triangle Theorem to find the length of the shorter leg. Then use that value to find the length of the longer leg. Shorter leg hypotenuse = 2 · shorter leg Longer leg longer leg = shorter leg ·3 8 = 2 · xy = 4 · 3 4 = xy = 43

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Example 5 Find Leg Lengths ANSWER The length of the shorter leg is 4. The length of the longer leg is 4. 3

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Now You Find Leg Lengths ANSWER 3.5 Find the value of each variable. Round your answer to the nearest tenth. 4. 5. ANSWER x = 21 ; y = 21 ≈ 36.4 3

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