Question

The difference between the area of a square and that of an equilateral triangle on the same base is $\frac{1}{4}c{m}^2$. What is the length of the side of the triangle?

• A. ${(4-\sqrt{3})}^{\frac{1}{2}}cm$
• B. ${(4+\sqrt{3})}^{\frac{1}{2}}cm$
• C. ${(4-\sqrt{3})}^{-\frac{1}{2}}cm$
• D. ${(4+\sqrt{3})}^{-\frac{1}{2}}cm$

Answer 1

The correct option is C ${(4-\sqrt{3})}^{-\frac{1}{2}}cm$

Let the base of the square and the equilateral triangle be $x$ cm.

Given, Area of square $-$ Area of equilateral triangle $=\frac{1}{4}$
$\Rightarrow x^2-\frac{\sqrt{3}}{4}{x}^2=\frac{1}{4}$

$\Rightarrow \frac{4x^2-\sqrt{3}{x}^2}{4}=\frac{1}{4}$
$\Rightarrow x^2(4-\sqrt{3})=1$
$\Rightarrow x=\sqrt{\frac{1}{4-\sqrt{3}}}$

$\Rightarrow x=(4-\sqrt{3}{)}^{-\frac{1}{2}}$