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Printjmc
prealgebra senior
Problem
What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 8 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form.
Solution
Drawing an altitude of an equilateral triangle splits it into two 30-60-90 right triangles:
The altitude is the longer leg of each 30-60-90 triangle, and the hypotenuse of each 30-60-90 triangle is a side of the equilateral triangle, so the altitude's length is times the side length of the triangle.
Therefore, the altitude of the equilateral triangle in the problem is , so the area of the equilateral triangle is . The perimeter of the triangle is . Thus, the ratio of area to perimeter is
The altitude is the longer leg of each 30-60-90 triangle, and the hypotenuse of each 30-60-90 triangle is a side of the equilateral triangle, so the altitude's length is times the side length of the triangle.
Therefore, the altitude of the equilateral triangle in the problem is , so the area of the equilateral triangle is . The perimeter of the triangle is . Thus, the ratio of area to perimeter is
Final answer
\frac{2\sqrt{3}}{3}