A triangle with angle measures 30°, 60°, and 90° has a perimeter of . What is the length of the

Question

A triangle with angle measures 30&deg;, 60&deg;, and 90&deg; has a perimeter of . What is the length of the longest side of the triangle?

Accepted Answer

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

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