If $A(-3, 5)$, $B(7, 12)$, $C(5, 3)$ and $D$ are the four vertices of parallelogram $ABCD$, what are the coordinates of point $D$?
To find the coordinates of $D$, I need to use the fact that opposite sides of a parallelogram are parallel and have the same length. This means that $AD$ and $BC$ are parallel and have the same length, and so are $AB$ and $CD$. One way to use this fact is to find the midpoint of $AC$ and $BD$, and then use the fact that the diagonals of a parallelogram bisect each other. Let me label the midpoint of $AC$ as $E$ and the midpoint of $BD$ as $F$. To find the coordinates of $E$, I can use the midpoint formula, which says that the midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ has coordinates $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. So, applying this formula to $AC$, I get that $E$ has coordinates $\left(\frac{-3 + 5}{2}, \frac{5 + 3}{2}\right) = (1, 4)$. Similarly, applying this formula to $BD$, I get that $F$ has coordinates $\left(\frac{7 + x}{2}, \frac{12 + y}{2}\right)$, where $x$ and $y$ are the unknown coordinates of $D$. Now, since the diagonals of a parallelogram bisect each other, I know that $E$ and $F$ are the same point. This means that their coordinates are equal, so I can set them equal to each other and solve for $x$ and $y$. I get that $1 = \frac{7 + x}{2}$ and $4 = \frac{12 + y}{2}$. Solving for $x$, I get that $x = -5$. Solving for $y$, I get that $y = -4$. Therefore, the coordinates of $D$ are $(-5, -4)$. 