Proof 5.15.

Using rectangular coordinates, prove that if the diagonals of a parallelogram are congruent, the parallelogram is a rectangle.
Proof: Consider the parallelogram . Denote the points of the parallelogram as follows: and where is some constant. Assume the diagonals of the parallelogram are congruent. This means that the lengths of Using the distance formula, we can determine the length using the coordinates of and . Notice, and . Since these distances are congruent, we can set them equal to one another as follows: . Since , the vertices of the parallelogram can be denoted as and . Notice that these points create a rectangle. Therefore, we can conclude that the diagonals of a parallelogram are congruent if the parallelogram is a rectangle.