Answer
see explanations.
Work Step by Step
We need to show that the opposite sides are parallel, adjacent sides are perpendicular, and each side is equal.
Step 1. Let A, B, C, D be the vertices $A(0,0),B(1,3),C(4,2),D(3,-1)$
Step 2. Find slopes of each side: $m_{AB}=\frac{0-3}{0-1}=3$, $m_{CD}=\frac{-1-2}{3-4}=3$, $m_{AD}=\frac{-1-0}{3-0}=-\frac{1}{3}$, $m_{BC}=\frac{3-2}{1-4}=-\frac{1}{3}$,
Step 3. Since $AB\parallel CD$ and $AD\parallel BC$, we have a parallelogram.
Step 4. Since $AB\perp AD$ and $CD\perp BC$, we have a rectangle.
Step 5. Using the distance formula, we can confirm that each side $AB, BC, CD, DA$ is equal to $\sqrt {10}$
Step 6. Thus we have a square.
