Exercise 27 Page 373

To find the coordinates of the missing vertex $F,$ we will follow four steps.

1. Plot the given vertices and draw two sides of the parallelogram.
2. Find the slope of one of the sides.
3. Use the rise and run of the obtained slope to find the missing vertex.
4. Verify that the sides are parallel.

Let's do it!

Plotting the Vertices

We know three of the vertices: $D(0,2),$ $E(\text{-}1,5),$ and $G(4,0).$ Let's plot them on a coordinate plane and draw two sides of the parallelogram.

Finding the Slope of $\overline{DE}$

With our graph, we can find the slope of $\overline{DE}.$

Using the observed changes in $x$ and $y,$ we can write the slope of $\overline{DE}.$

Finding the Missing Vertex

By definition, opposite sides of a parallelogram are parallel. Therefore, the slope of $\overline{GF}$ will be the same as the slope of $\overline{DE}.$ Let's start at $G$ and use the rise and run from the previous step to find vertex $F.$

The missing vertex $F$ occurs at the point $(3,3).$ Let's draw the parallelogram.

Verifying that the Sides are Parallel

By construction, we already know that $\overline{DE}\parallel \overline{GF}.$ However, for us to be certain that our quadrilateral is a parallelogram, we need both pairs of opposite sides to have equal slopes. Let's calculate the slopes of $\overline{DG}$ and $\overline{EF},$ and check that they are the same.

The slopes of $\overline{DG}$ and $\overline{EF}$ are equal! Because the opposite sides are parallel, the polygon is a parallelogram.