We know three of the vertices: D(-4,-2), E(-3,1), and F(3,3). Let's plot them on a coordinate plane and draw two sides of the parallelogram.
Finding the Slope of EF
With our graph, we can find the slope of EF.
Using the observed changes in x and y, we can write the slope of EF.
Slope=Rise/Run ⇔ Slope= 2/6=1/3
Finding the Missing Vertex
By definition, opposite sides of a parallelogram are parallel. Therefore, the slope of EF will be the same as the slope of DG. Let's start at D and use the rise and run from the previous step to find vertex G.
The missing vertex G occurs at the point (2,0). Let's draw the parallelogram.
Verifying that the Sides are Parallel
By construction, we already know that DG ∥ EF. 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 DE and GF, and check that they are the same.
The slopes of DE and GF are equal!
Slope=Rise/Run ⇔ Slope= 3/1=3
Because the opposite sides are parallel, the polygon is a parallelogram.
