Question

If the centroid of a triangle is at $(1,3)$ and two of its vertices are $(-7,6)$ and $(8,5)$, then find third vertex of the triangle

Answer 1

Given, $A(-7,6),B(8,5)$
and centroid $(1,3)$.

the third vertex be $(x,y)$.

By applying centroid's formula

$\Rightarrow (\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})=(1,3)$

$\Rightarrow (\frac{-7+8+x}{3},\frac{6+5+y}{3})=(1,3)$

$\Rightarrow (\frac{1+x}{3},\frac{11+y}{3})=(1,3)$

$\Rightarrow \frac{1+x}{3}=1,\frac{11+y}{3}=3$

$\Rightarrow 1+x=3,11+y=9$

$\Rightarrow x=3-1,y=9-11$

$\Rightarrow x=2,y=-2$

The third vertex is $(2,-2)$.