The coordinates of the centroid of a triangle ABC are $(-1,-2)$. What are the coordinates of vertex C, if the coordinates of A and B are $(6,-4)$ and $(-2,2)$ respectively?

Let the coordinates of vertex C be $(x_3,y_3)$.Given:$A=(x_1,y_1)=(6,-4)$$B=(x_2,y_2)=(-2,2)$$\text{Centroid }G=(x,y)=(-1,-2)$ The formula for the coordinates of the centroid of a triangle is:
$$x=\frac{x_1+x_2+x_3}{3},y=\frac{y_1+y_2+y_3}{3}$$
Substituting the given values into the x-coordinate formula:
$$-1=\frac{6+(-2)+x_3}{3}$$
$$-3=4+x_3$$
$$x_3=-3-4=-7$$
Substituting the given values into the y-coordinate formula:
$$-2=\frac{-4+2+y_3}{3}$$
$$-6=-2+y_3$$
$$y_3=-6+2=-4$$
Therefore, the coordinates of vertex C are $(-7,-4)$.