Question

From a point in the interior of an equilateral triangle the perpendicular distances of the sides are $\sqrt{3}cm,2\sqrt{3}cmand5\sqrt{3}cm$. What is the perimeter (in cm) of the triangle?

• A. $45$ cm
• B. $56$ cm
• C. $42$ cm
• D. $48$ cm

Answer 1

The correct option is D $48$ cm

Let each side of equilateral $△ABC$ be ${}^{\prime }{a}^{\prime }cm$

In $△ABC,OD=\sqrt{3}cm$

$OE=2\sqrt{3}cm$

$OF=5\sqrt{3}cm$

Now, $ar△ABC=ar△BOC+ar△COA+ar△AOB$

$\Rightarrow \frac{\sqrt{3}}{4}{a}^2=\frac{1}{2}[BC\times OD+AC\times DE+AB\times OF]=\frac{1}{2}[a\times \sqrt{3}+a\times 2\sqrt{3}+a\times 5\sqrt{3}]$

$\Rightarrow \frac{\sqrt{3}}{4}{a}^2=\frac{1}{2}\times 8a\sqrt{3}$

$\Rightarrow a=16cm$

$\therefore$Perimeter of $△ABC=3a=3\times 16=48cm$