The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k 3 centimeters, where k

Question

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 ?

Accepted Answer

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 /mo> . 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 60 /mo> 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 30 /mo> angle has a length of k and each hypotenuse has a length of 2 k . 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 2 k . 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, 2 k + 2 k + 2 k = 624 , or 6 k = 624 . Dividing both sides of this equation by 6 yields k = 104 .

Question the-pe