The coordinates of the vertices for the figure HIJK are H(0, 5), I(3, 3), J(4, –1), and K(1, 1).

To determine if it is a parallelogram, use the converse of the parallelogram diagonal theorem. This states that if the diagonals
, then the quadrilateral is a parallelogram.

The midpoint of HJ is
and the midpoint of IK is (2, 2).

Therefore, HIJK is a parallelogram because the diagonals
, which means they bisect each other.

well, HIJK is a parallelogram only if its diagonals bisect each other, if that's so, the midpoint of HJ is the same as the midpoint of IK, let's check

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ H(\stackrel{x_1}{0}~,~\stackrel{y_1}{5})\qquad J(\stackrel{x_2}{4}~,~\stackrel{y_2}{-1}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 4 + 0}{2}~~~ ,~~~ \cfrac{ -1 + 5}{2} \right)\implies \left( \cfrac{4}{2}~~,~~\cfrac{4}{2} \right)\implies \boxed{(2~~,~~2)} \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ I(\stackrel{x_1}{3}~,~\stackrel{y_1}{3})\qquad K(\stackrel{x_2}{1}~,~\stackrel{y_2}{1}) ~\hfill \left(\cfrac{ 1 + 3}{2}~~~ ,~~~ \cfrac{ 1 + 3}{2} \right)\implies \boxed{(2~~,~~2)}[/tex]