Question

Two vertices of a triangle are $(3,-5)$ and $(-7,4)$. If its centroid is$(2,-1)$, then find the third vertex.

Answer 1

Finding the third vertex of the triangle:

Let $A$ $(x_1,y_1)$ be the first vertex of the triangle having coordinates$(3,-5)$,

$B$ $(x_2,y_2)$be the second vertex of the triangle having coordinates$(-7,4)$ and

$C$ be the third vertex of the triangle having coordinates $(x_3,y_3)$

The coordinates of the centroid of a triangle $=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

Therefore,$(2,-1)$ $=\left(\frac{3+(-7)+x_3}{3},\frac{-5+4+y_3}{3}\right)$

On equating x and y coordinates on both sides

$2=\frac{-4+x_3}{3}$ and $-1=\frac{-1+y_3}{3}$

$\Rightarrow 6+4=x_3$ and $-3+1=y_3$

$\Rightarrow x_3=10$ and $-2=y_3$

Hence, the coordinates of the third vertex $C(x_3,y_3)$ is$(10,-2)$.