Answer
$AB\parallel CD$ and $BC\parallel DA$
Work Step by Step
Determine the slope of the line passing through $A(1,1)$ and $B(7,4)$:
$$m_{AB}=\dfrac{y_B-y_A}{x_B-x_A}=\dfrac{4-1}{7-1}=\dfrac{1}{2}.$$
Determine the slope of the line passing through $B(7,4)$ and $C(5,10)$:
$$m_{BC}=\dfrac{y_C-y_B}{x_C-x_B}=\dfrac{10-4}{5-7}=-3.$$
Determine the slope of the line passing through $C(5,10)$ and $D(-1,7)$:
$$m_{CD}=\dfrac{y_D-y_C}{x_D-x_C}=\dfrac{7-10}{-1-5}=\dfrac{1}{2}.$$
Determine the slope of the line passing through $D(-1,7)$ and $A1,1)$:
$$m_{DA}=\dfrac{y_A-y_D}{x_A-x_D}=\dfrac{1-7}{1-(-1)}=-3.$$
Because $m_{AB}=m_{CD}$, it means that $AB$ is parallel to $CD$.
Because $m_{BC}=m_{DA}$, it means that $BC$ is parallel to $DA$.
From $AB\parallel CD$ and $BC\parallel DA$ it follows that $ABCD$ is a parallelogram
