Answer
The diagonals are perpendicular and have the same length, so the given points form a square.
Work Step by Step
Plot the points and note that the diagonals connect
$1.$
$A=(0,0)$ and $C=(4,2)$
$m_{AC}=\displaystyle \frac{2-0}{4-0}=\frac{1}{2}$
$d(A,B)=\sqrt{(4-0)^{2}+(2-0)^{2}}=\sqrt{20}=2\sqrt{5}$
$2.$
$B=(3,-1)$ and $D(1,3)$
$m_{BD}=\displaystyle \frac{3-(-1)}{1-3}=-2$
$d(B,D)=\sqrt{(1-3)^{2}+(3-(-1))^{2}}=\sqrt{20}=2\sqrt{5}$
$m_{AC}\cdot m_{BD}=-1$ , so the diagonals are perpendicular,
so ABCD is a rhombus.
They also have the same length,
so ABCD is a rectangle.
A rhombus that is a rectangle is called a square.
