Exercise 45 Page 374

Let's begin by plotting the given points in the coordinate plane. First, we will consider the case when $\overline{XY}$ is a side of the drawn parallelogram.

Since the opposite sides of a parallelogram are parallel, they have the same slope. Therefore, the slope between $X$ and $Y$ should be the same as the slope between $W$ and $Z.$ Let's evaluate the slope of the segment $XY.$

As we are moving $3$ units in the positive horizontal direction, we are also moving $2$ units in the negative vertical direction. With this, we can find point $Z.$ Starting in point $W,$ let's move $3$ units to the right and $2$ units down.

The possible coordinates of vertex $Z$ are $(4,0).$ However, vertex $Z$ can also occur on the other side of $W.$ To find the coordinates of the second possible position of $Z,$ we will move $3$ units to the left and $2$ units up.

The second possible coordinates of vertex $Z$ are $(\text{-}2,4).$ Now, we will consider the case when segment $XW$ is a side of this parallelogram. This means the slope between $W$ and $X$ should be the same as the slope between $Y$ and $Z.$ Let's evaluate the slope of the segment $WX.$

As we are moving $2$ units in the positive horizontal direction, we are also moving $4$ units in the positive vertical direction. With this, we can find point $Z.$ Starting in point $Y,$ let's move $2$ units to the right and $4$ units up.

The third possible coordinates of the fourth vertex $Z$ are $(8,8).$