We know three of the vertices: D(-2,-4), F(0,7), and G(1,0). Let's plot them on a coordinate plane and draw two sides of the parallelogram.
Finding the Slope of GF
With our graph, we can find the slope of GF.
Using the observed changes in x and y, we can write the slope of GF.
Slope=Rise/Run ⇔ Slope= 7/-1=-7
Finding the Missing Vertex
By definition, opposite sides of a parallelogram are parallel. Therefore, the slope of GF will be the same as the slope of DE. Let's start at D and use the rise and run from the previous step to find vertex E.
The missing vertex E occurs at the point (-3,3). Let's draw the parallelogram.
Verifying that the Sides are Parallel
By construction, we already know that DE ∥ 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 DG and EF, and check that they are the same.
The slopes of DG and EF are equal!
Slope=Rise/Run ⇔ Slope= 4/3
Because the opposite sides are parallel, the polygon is a parallelogram.
