Question

$P$ is point in the interior of an equilateral triangle with side $a$ units. If $P_1$, $P_2$, $P_3$ are the distances of $P$ from the three sides of the triangle, then $P_1+P_2+P_3$:

• A. Equals $2a/3$ units
• B. Equals $\frac{a\sqrt{3}}{2}$ units
• C. Is more than $a$ units
• D. Cannot be determined unless the location of $P$ is specified

Answer 1

The correct option is B Equals $\frac{a\sqrt{3}}{2}$ units

Let the point $P$ be incenter/centroid of an equilateral triangle

The length of side of an equilateral triangle is $a$

Then we have $P_1=P_2=P_3=\frac{a}{2\sqrt{3}}$

Therefore $P_1+P_2+P_3=\frac{\sqrt{3}a}{2}$

So the correct option is $B$