OG详解-OG10 数学2 Q27

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 60° 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° 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.