# Geometry Puzzle – Challenge 76

This is a perfect math challenge for those who enjoy solving complicated mathematics and critical thinking challenges. Let’s challenge your brain!

## Challenge:

If the perimeter of an equilateral triangle is 2x meters and its area is x square meters, then what is the length of one side of the triangle in meters?

A- $$\sqrt{3}$$

B- $$\frac{\sqrt{3}}{2}$$

C- $$2\sqrt{3}$$

D- $$\frac{2\sqrt{3}}{3}$$

E- 3

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The perimeter of the equilateral triangle is 2x meters. So, one side is $$\frac{2}{3}x$$ meters.
The area of an equilateral triangle $$= \frac{s^2 \sqrt{3}}{4}$$ (s is one side of the triangle)
The perimeter of the triangle is twice its area. So:
$$2x = 2 (\frac{s^2 \sqrt{3}}{4}) → 2x = (\frac{s^2 \sqrt{3}}{2})$$
Replace the s with $$\frac{2}{3}x$$. Then:
$$2x = \frac{(\frac{2}{3} x)^2 \sqrt{3}}{2} = \frac{\frac{4}{9} x^2 \sqrt{3}}{2 }→ 4x = \frac{4}{9} x^2 \sqrt{3} → 4 = \frac{4}{9} x\sqrt{3} → 9 = x\sqrt{3}→ \frac{9}{\sqrt{3} }= x → \frac{9}{\sqrt{3} } × \frac{\sqrt{3}}{\sqrt{3} } = x → x = 3\sqrt{3}$$
Then, one side of the triangle is: $$\frac{2}{3}x =\frac{ 2}{3}(3\sqrt{3}) = 2\sqrt{3}$$

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