Points $A(5,3)$,$B(-2,3)$,and $D(5,-4)$ are three vertices of a square $ABCD$. Plot these points on a graph paper and hence find the coordinates of the vertex $C$.

(C) $1$. Plot the points $A(5,3)$,$B(-2,3)$,and $D(5,-4)$ on the Cartesian plane.
$2$. Join the points $AB$ and $AD$. Since $ABCD$ is a square,all its sides are equal in length and each interior angle is $90^{\circ}$.
$3$. The side $AB$ is horizontal (parallel to the $x$-axis) with length $|5 - (-2)| = 7$ units. The side $AD$ is vertical (parallel to the $y$-axis) with length $|3 - (-4)| = 7$ units.
$4$. To complete the square,the vertex $C$ must have the same $x$-coordinate as $B$ (which is $-2$) and the same $y$-coordinate as $D$ (which is $-4$).
$5$. Therefore,the coordinates of the vertex $C$ are $(-2,-4)$.