Question

Two vertices of a triangle are $(-1,4)$ and $(5,2)$. If the centroid is $(0,-3)$ 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 a triangle having coordinates$(-1,4)$, $B(x_2,y_2)$be the second vertex of a triangle having coordinates $(5,2)$ and

$C$ be the third vertex of a triangle having coordinates $\left(x_3,y_3\right)$

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

Therefore, $\left(0,-3\right)=\left(\frac{-1+5+x_3}{3},\frac{4+2+y_3}{3}\right)$

Equating $x$ and $y$-coordinates on both sides,

$0=\frac{4+x_3}{3}$ and $-3=\frac{6+y_3}{3}$

$\Rightarrow$ $-4=x_3$ and $-9-6=y_3$

$-15=y_3$

Hence, the coordinates of the third vertex $\mathrm{C}\left(x_3,y_3\right)$ are$(-4,-15)$.