Question

which statement proves that pqrs is a parallelogram? the midpoint of rp is (4, 5 1/2) and the slope of rp is -9/2. the slopes of sp and rq are both -2 and sp = rq = √45. the slopes of rs and qp are both 3 and sp = rq = √45. the midpoint of sq is (4, 5 1/2) and sq = 5.

Explanation:

Step1: Recall parallelogram properties

A quadrilateral is a parallelogram if one - pair of opposite sides are parallel and congruent. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$ and the distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Analyze the first option

The mid - point and slope of a single diagonal ($\overline{RP}$) do not prove that $PQRS$ is a parallelogram.

Step3: Analyze the second option

The slopes of $\overline{SP}$ and $\overline{RQ}$ being equal shows they are parallel. Calculating the length of $SP$ using the distance formula with $S(2,7)$ and $P(5,1)$: $SP=\sqrt{(5 - 2)^2+(1 - 7)^2}=\sqrt{9 + 36}=\sqrt{45}$. Calculating the length of $RQ$ with $R(3,10)$ and $Q(6,4)$: $RQ=\sqrt{(6 - 3)^2+(4 - 10)^2}=\sqrt{9+36}=\sqrt{45}$. Since one pair of opposite sides are parallel and congruent, this proves the quadrilateral is a parallelogram.

Step4: Analyze the third option

The slopes of $\overline{RS}$ and $\overline{QP}$ being equal is correct for parallel sides, but the equal lengths given are for non - opposite sides ($SP$ and $RQ$ instead of $RS$ and $QP$), so this does not prove it is a parallelogram.

Step5: Analyze the fourth option

The mid - point and length of a single diagonal ($\overline{SQ}$) do not prove that $PQRS$ is a parallelogram.

Answer:

The slopes of $\overline{SP}$ and $\overline{RQ}$ are both $- 2$ and $SP = RQ=\sqrt{45}$.