Answer
Find slopes of the lines containing pairs of points
$P_{1}=(1,-1),P_{2}=(4,1),P_{3}=(2,2),P_{4}=(5,4)$
$m_{12}=\displaystyle \frac{1-(-1)}{4-1}=\frac{2}{3}$
$m_{13}=\displaystyle \frac{2-(-1)}{2-1}=3$
$m_{24}=\displaystyle \frac{4-1}{5-4}=3$
$m_{34}=\displaystyle \frac{4-2}{5-2}=\frac{2}{3}$
Criteria for Parallel Lines
Two nonvertical lines are parallel if and only if their slopes are equal and they have different $y$ -intercepts.
We see that $m_{13}=m_{24}$, and $ m_{12}=m_{34}$
So,
the sides $\overline{P_{1}P_{3}}$ and $\overline{P_{2}P_{4}}$ are parallel, and
the sides $\overline{P_{1}P_{2}}$ and $\overline{P_{3}P_{4}}$ are parallel.
Therefore, the given points are vertices of a parallelogram $P_{1}P_{2}P_{4}P_{3}.$
Work Step by Step
All steps are given in the answer.
