Question

(a) find the length of $overline{pq}$ and the length of $overline{rs}$. give exact answers (not decimal approximations). length of $overline{pq}$: $2sqrt{10}$ length of $overline{rs}$: $2sqrt{10}$ (b) find the slope of $overline{pq}$ and the slope of $overline{rs}$. slope of $overline{pq}$: $\frac{1}{3}$ slope of $overline{rs}$: $\frac{1}{3}$ (c) from parts (a) and (b), what can we conclude? the quadrilateral is a parallelogram because it has one pair of opposite sides that are congruent, even though those sides are not parallel. the quadrilateral is a parallelogram because it has one pair of opposite sides that are both congruent and parallel. the quadrilateral is a parallelogram because it has one pair of opposite sides that are parallel, even though these sides are not congruent. the quadrilateral is not a parallelogram.

Explanation:

A quadrilateral is defined as a parallelogram if one pair of opposite sides is both congruent (equal in length) and parallel (equal in slope). From parts (a) and (b), $\overline{PQ}$ and $\overline{RS}$ are congruent (same length $2\sqrt{10}$) and parallel (same slope $\frac{1}{3}$).

Answer:

The quadrilateral is a parallelogram because it has one pair of opposite sides that are both congruent and parallel.