Question

If centroid of a triangle having two of its vertices $A(1,3),B(2,4)$ is $(0,0)$. Then the third vertex is _______ .

Answer 1

Let the coordinates of the third vertex be $(x,y).$

Centroid of a triangle having vertices $A(x_1,y_1),B(x_2,y_2)$ and $C(x_3,y_3)$ is given by,

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

Given, Centroid of the triangle is $(0,0).$

Then, $(0,0)=(\frac{1+2+x}{3},\frac{3+4+y}{3})$

$⟹(0,0)=(\frac{3+x}{3},\frac{7+y}{3})$

$⟹\frac{x+3}{3}=0$ and $\frac{y+7}{3}=0$

$⟹x=-3$ and $y=-7$

Hence, coordinates of third vertex is $(-3,-7).$