Question

A point $P$ is inside an equilateral triangle.If the distances of $P$ from the sides are $4,5,6$units, the length of the side of the triangle is

• A. $5\sqrt{3}$
• B. $10\sqrt{3}$
• C. $15\sqrt{3}$
• D. $20\sqrt{3}$

Answer 1

The correct option is B $10\sqrt{3}$

$\because △ABC=△PBC+△PCA+△PAB$
Area of $△ABC=$Area of $△PBC+$Area of $△PCA+$Area of $△PAB$
$\Rightarrow \frac{a^2\sqrt{3}}{4}=\frac{1}{2}.a.4+\frac{1}{2}.a.5+\frac{1}{2}.a.6$
$\Rightarrow \frac{a^2\sqrt{3}}{4}=\frac{1}{2}[4a+5a+6a]$
$\Rightarrow \frac{a^2\sqrt{3}}{4}=\frac{1}{2}\times 15a$
$\Rightarrow a=\frac{30}{\sqrt{3}}=10\sqrt{3}$unit.