Bell Ringer

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

You can use the Pythagorean Theorem to find the value of b. Example 1 In the diagram, PQR is a 30° –60° –90° triangle with PQ = 2 and PR = 1. Find the value of b. 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. 12 + b2 = 22 Substitute. 1 + b2 = 4 Simplify. b2 = 3 Subtract 1 from each side. Take the square root of each side. b = 3 3

hypotenuse = 2 · shorter leg Example 2 In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse. 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. ANSWER The length of the hypotenuse is 24. 4

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

Find the value of x. Write your answer in radical form. 1. Now You Try  Find Lengths in a Triangle Find the value of x. Write your answer in radical form. 1. ANSWER 14 2. ANSWER 3 3. ANSWER 10 3

longer leg = shorter leg · 3 Example 4 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. Find Shorter Leg Length SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times 3 . 30° –60° –90° Triangle Theorem longer leg = shorter leg · 3 Substitute. 5 = x · 3 = x 5 3 Divide each side by . 2.9 ≈ x Use a calculator. ANSWER The length of the shorter leg is about 2.9. 7

hypotenuse = 2 · shorter leg Longer leg longer leg = shorter leg · 3 Example 5 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. Find Leg Lengths 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 · x y = 4 · 3 4 = x y = 4 3 8

The length of the shorter leg is 4. Example 5 Find Leg Lengths ANSWER The length of the shorter leg is 4. The length of the longer leg is 4 . 3 9

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

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