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

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 . If x is the measure of the shortest leg, then the measure of the other leg is and the measure of the hypotenuse is 2x. 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 2x, or 2(6), which is 12.