2. Given a quadrilateral with vertices (−1, 3), (1, 5), (5, 1), and (3,−1):

a. Prove that quadrilateral is a rectangle.

In every rectangle, the two diagonals have the same length. If a quadrilateral's diagonals have the same length, that doesn't mean it has to be a rectangle, but if a parallelogram's diagonals have the same length, then it's definitely a rectangle.

So first of all, let's prove this is a parallelogram. The basic definition of a parallelogram is that it is a quadrilateral where both pairs of opposite sides are parallel.

So let's name the vertices as:

First pair of opposite sides:

Slope:

Second pair of opposite sides:

Slope:

So in fact this is a parallelogram. The other thing we need to prove is that the diagonals measure the same. Using distance formula:

So the diagonals measure the same, therefore this is a rectangle.

noncoplanar points are points that do not lie on the same line.