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Bell Ringer. 30-60-90 Triangles. A Right Triangle with angle measures of 30, 60, and 90 are called 30-60-90 triangles. Example 1. In the diagram, PQR is a 30 ° – 60 ° – 90 ° triangle with PQ = 2 and PR = 1 . Find the value of b. 3. b =. Take the square root of each side.
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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.
Example 1 In the diagram, PQR is a 30°–60°–90°triangle with PQ = 2 and PR = 1. Find the value of b. 3 b = Take the square root of each side. 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.
Example 2 Find Hypotenuse Length In the 30°–60°–90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse. 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. The length of the hypotenuse is 24. ANSWER
Example 3 SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times . longer leg = shorter leg · 30° –60°–90° Triangle Theorem = 5 · Substitute. 3 3 3 3 The length of the longer leg is 5 . ANSWER 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.
Now You Try 3 ANSWER 3 3 10 ANSWER Find Lengths in a Triangle Find the value of x. Write your answer in radical form. 1. 14 ANSWER 2. 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. SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times . longer leg = shorter leg · 30°–60°–90° Triangle Theorem 5 = x· Substitute. 3 3 3 3 5 = x Divide each side by . 3 Find Shorter Leg Length The length of the shorter leg is about 2.9. ANSWER 2.9 ≈ x Use a calculator.
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. Shorter leg Longer leg 3 3 3 longer leg = shorter leg · hypotenuse = 2 · shorter leg y = 4 · y = 4 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. 8 = 2 ·x 4 = x
Example 5 The length of the shorter leg is 4. ANSWER The length of the longer leg is 4 . 3 Find Leg Lengths
Now You 3 x = 21;y = 21 ≈ 36.4 ANSWER Find Leg Lengths Find the value of each variable. Round your answer to the nearest tenth. 4. 3.5 ANSWER 5.