# Proof 5.15

Using rectangular coordinates, prove that if the diagonals of a parallelogram are congruent, the parallelogram is a rectangle.
Consider the parallelogram ABCD. Let the points of the parallelogram be denoted as  and  where c is any constant. Create the diagonals  and . Using the distance formula () we can determine the length of the diagonals. The distance of  is given by  and the distance of  is given by . Since , we know that the diagonals of a parallelogram are not congruent. Now consider the rectangle EFGH. Let the points of the rectangle be denoted as  and . Create the diagonals  and . Using the Pythagorean Theorem, which we proved earlier, we can determine the length of the diagonals. Consider  where  has a distance of  because the y coordinates are the same and it only changes horizontally.  has a distance of  because the x coordinates are the same and it only changes vertically. Based on this information, the length of the hypotenuse of  is given by  so . Consider  has a distance that is given by  because it does not change vertically.  has a distance given by  because it only changes horizontally. The distance of the hypotenuse is given by  so . Based on this information, we can conclude that the diagonals of a rectangle are congruent. Since we previously determined that this is not true for a parallelogram, we know that diagonals of a parallelogram are congruent if the parallelogram is a rectangle.