The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k 3 centimeters, where k is a constant. What is the value of k?

Explanation

The correct answer is 104. An equilateral triangle is a triangle in which all three sides have the same length and all three angles have a measure of 60°. The height of the triangle, k 3, is the length of the altitude from one vertex. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the 60o angle in each 30-60-90 right triangle. Since the altitude has a length of k 3, it follows from the properties of 30-60-90 right triangles that the side across from each o30 angle has a length of k and each hypotenuse has a length of 2k. In this case, the hypotenuse of each 30-60-90 right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral triangle is 2k. The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral triangle is 624; therefore, 2k + 2k + 2k = 624 or 6k = 624. Dividing both sides of this equation by 6 yields k =104.