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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

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

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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 $60 degree$ . The height of the triangle, $k StartRoot 3 EndRoot$ , 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 $60 degree$ angle in each 30-60-90 right triangle. Since the altitude has a length of $k StartRoot 3 EndRoot$ , it follows from the properties of 30-60-90 right triangles that the side across from each $30 degree$ 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 $6 k equals 624$ . Dividing both sides of this equation by $6$ yields $k equals 104$ .