Question

From a point within an equilateral triangle perpendiculars drawn to the three sides are $6cm,7cm$ and $8cm$, respectively the length of the side of the triangle is

• A. $7cm$
• B. $10.5cm$
• C. $14\sqrt{3}cm$
• D. $\frac{14\sqrt{3}}{3}$cm

Answer 1

The correct option is C $14\sqrt{3}cm$

Let the length of each side of the $\Delta$ be $x$ cm.
$\Rightarrow \frac{1}{2}\times x\times 6+\frac{1}{2}\times x\times 7+\frac{1}{2}x\times 8=\frac{\sqrt{3}}{4}{x}^2$
$\Rightarrow x(3+3.5+4)=\frac{\sqrt{3}}{4}{x}^2$
$\Rightarrow x=\frac{10.5\times 4}{\sqrt{3}}=\frac{42}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{42\sqrt{3}}{3}=14\sqrt{3}cm$