Let $A(1,0), B(6,2)$ and $C\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $A B C$. If $P

Question

Let $A(1,0), B(6,2)$ and $C\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $A B C$. If $P$ is a point inside the triangle $A B C$ such that the triangles $A P C, A P B$ and $B P C$ have equal areas, then the length of the line segemnt $P Q$, where $Q$ is the point $\left(-\frac{7}{6},-\frac{1}{3}\right)$, is

Professor PC Thomas Classes · Answer

(5) : Since, $\operatorname{ar}(	riangle P A B)=\operatorname{ar}(	riangle P A C)$\ $=\operatorname{ar}(	riangle P B C)$ [Given]\ $	herefore \quad P$ will be centroid of $	riangle A B C$.\ So, coordinates of $P$ are $\left(\frac{17}{6}, \frac{8}{3}ight)$. Now, $P Q=\sqrt{\left(\frac{24}{6}ight)^{2}+\left(\frac{9}{3}ight)^{2}}=5$\