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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

A triangle with angle measures 30°, 60°, and 90° has a perimeter of $18 plus 6, times, the square root of 3$ . What is the length of the longest side of the triangle?

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Explanation

The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a special right triangle. The side measures of this type of special triangle are in the ratio $2 to 1 to the square root of 3$ . If x is the measure of the shortest leg, then the measure of the other leg is $the square root of 3 end root x$ and the measure of the hypotenuse is 2x. The perimeter of the triangle is given to be $18, plus 6 times, the square root of 3, end root$ , and so the equation for the perimeter can be written as $2 x, plus x, plus the square root of 3 end root x, equals 18, plus 6 times, the square root of 3$ . Combining like terms and factoring out a common factor of x on the left-hand side of the equation gives $open parenthesis, 3 plus the square root of 3, close parenthesis, times x, equals 18, plus 6 times the square root of 3$ . Rewriting the right-hand side of the equation by factoring out 6 gives $open parenthesis, 3 plus the square root of 3, close parenthesis, times x, equals 6 times, open parenthesis, 3 plus the square root of 3, close parenthesis$ . Dividing both sides of the equation by the common factor $open parenthesis, 3 plus the square root of 3, close parenthesis$ gives x = 6. The longest side of the right triangle, the hypotenuse, has a length of 2x, or 2(6), which is 12.