If the origin is the centroid of a triangle $ABC$ having vertices $A(a, 1, 3)$,$B(-2, b, -5)$,and $C(4, 7, c)$,find the values of $a, b, c$.

(A) The vertices of triangle $ABC$ are $A(a, 1, 3)$,$B(-2, b, -5)$,and $C(4, 7, c)$.
The centroid $G$ of a triangle with vertices $(x_1, y_1, z_1)$,$(x_2, y_2, z_2)$,and $(x_3, y_3, z_3)$ is given by $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$.
Given that the centroid is the origin $G(0, 0, 0)$,we have:
$G(0, 0, 0) = \left(\frac{a-2+4}{3}, \frac{1+b+7}{3}, \frac{3-5+c}{3}\right)$.
Equating the coordinates:
$0 = \frac{a+2}{3} \implies a+2 = 0 \implies a = -2$.
$0 = \frac{b+8}{3} \implies b+8 = 0 \implies b = -8$.
$0 = \frac{c-2}{3} \implies c-2 = 0 \implies c = 2$.
Thus,the values are $a = -2, b = -8, c = 2$.