Question

The difference between the area of a square and that of an equilateral triangle on the same base is $\frac{1}{4}c{m}^3$ . What is the length of the side of the triangle?

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

Answer 1

The correct option is B $(4+\sqrt{3}{)}^{1/2}cm$

Let the side of the square and triangle be $=a$ m

Area of the square $=a^2$
Area of an equilateral triangle with length of side $=\frac{\sqrt{3}}{4}{a}^2$
Given, $a^2-\frac{\sqrt{3}}{4}{a}^2=\frac{1}{4}$
$\Rightarrow \left(\frac{4-\sqrt{3}}{4}\right)a^2=\frac{1}{4}$
$\Rightarrow a^2=\frac{1}{4-\sqrt{3}}=\frac{4+\sqrt{3}}{4-3}=4+\sqrt{3}$
$\Rightarrow a={(4+\sqrt{3})}^{\frac{1}{2}}$ cm