5. A point inside an equilateral triangle is 5, 7, and 8 units away from the three sides. What is the length of the side of the triangle?

(A) The data is insufficient

(B) The given configuration cannot exist

(C) The side length is 20

(D) The side length is $14 \sqrt{3}$

(E) The side length is $\frac{40}{3} \sqrt{3}$.
5. The answer is (E). Referring to the adjacent figure, the area of the equilateral triangle with unknown side length $l$ can be determined in two ways:
1) as the sum of the areas of three triangles with base $l$ and heights $5, 7, 8$
2) using the usual area formula, keeping in mind that the height of an equilateral triangle is $l \frac{\sqrt{3}}{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_9dd5da716b9e5be06201g-16.jpg?height=212&width=227&top_left_y=1954&top_left_x=1708)
Thus, $\frac{l \cdot(5+7+8)}{2}=\frac{1}{2} l^{2} \frac{\sqrt{3}}{2}$, from which $l=\frac{40}{3} \sqrt{3}$.