Question

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 bisec have different midpoints are perpendicular have the same midpoint

Explanation:

Step1: Recall parallelogram diagonal theorem

The converse of the parallelogram diagonal theorem states that if the diagonals of a quadrilateral bisect each other (i.e., have the same mid - point), then the quadrilateral is a parallelogram.

Step2: Calculate mid - point of HJ

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For points $H(0,5)$ and $J(4, - 1)$, we have $x=\frac{0 + 4}{2}=2$ and $y=\frac{5+( - 1)}{2}=2$. So the mid - point of $\overline{HJ}$ is $(2,2)$.

Step3: Analyze the result

The mid - point of $\overline{IK}$ is given as $(2,2)$ and the mid - point of $\overline{HJ}$ is $(2,2)$. Since the diagonals have the same mid - point, they bisect each other.

Answer:

1. The first blank: bisect each other
2. The second blank: $(2,2)$
3. The third blank: have the same midpoint