A square of maximum area is inscribed in an equilateral triangle. If the side of the triangle is equal to \((6+4\sqrt{3})\) cm, then what is the area of the square?
\(36\ \text{cm}^2\)
For an equilateral triangle of side \(a\), the side of the largest square inscribed with its base on one side of the triangle is \(x=a(2\sqrt{3}-3)\). With \(a=6+4\sqrt{3}\): \(x=(6+4\sqrt{3})(2\sqrt{3}-3)=12\sqrt{3}-18+24-12\sqrt{3}=6\) cm. So the area of the square \(=x^2=36\ \text{cm}^2\).
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