Geometry Puzzle – Challenge 76

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: \(\frac{2}{3}x =\frac{ 2}{3}(3\sqrt{3}) = 2\sqrt{3}\)