We know that three of the vertices are A(2,4), B(5,2), and C(3,- 1). Let's plot them on a coordinate plane and draw two of the sides of the parallelogram.
Finding the Slope of BA
Using our graph, we will find the slope of BA.
Using the observed changes in x and y, we can write the slope of BA.
Slope=Rise/Run ⇔ Slope= 2/- 3=- 2/3
Finding the Missing Vertex
By definition, opposite sides of a parallelogram are parallel. Therefore, the slope of CD will be the same as the slope of BA. Let's start at C and use the rise and run from the previous step to find vertex D.
The missing vertex D occurs at the point (0,1). Let's draw the parallelogram.
Verifying that the Sides are Parallel
By construction, we already know that BA ∥ CD. 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 DA and CB, and check that they are the same.
The slopes of DA and CB are equal!
Slope=Rise/Run ⇔ Slope= 3/2
Therefore, the opposite sides are parallel and the polygon is a parallelogram.
