# 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.

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