Question

A point $P$ is inside an equilateral triangle. If the distances of $P$ from the sides are $8,10,12$ units, the lengths of a side of the triangle is?

• A. $10$
• B. 10$\sqrt{3}$
• C. 20$\sqrt{3}$
• D. 30$\sqrt{3}$

Answer 1

The correct option is C 20$\sqrt{3}$

We can see that from point P line joining the vertices are dividing triangle into 3 Parts

And sum of Area of these 3 will be equal to Area of Full Triangle as follows:

$\frac{1}{2}a\ast 10+\frac{1}{2}a\ast 8+\frac{1}{2}a\ast 12=\frac{\sqrt{3}}{4}{a}^2\Rightarrow \frac{60}{\sqrt{3}}a=a^2\Rightarrow a=20\sqrt{3}$

Therefore, Answer is $C$