Question

which statement proves that pqrs is a parallelogram? 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 mid - point of rp is 4, 5 1/2 and the slope of rp is -9/2. the mid - point 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 is parallel and congruent.

Step2: Calculate slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For two points $S(2,7)$ and $P(5,1)$:
$m_{SP}=\frac{1 - 7}{5 - 2}=\frac{-6}{3}=-2$
For two points $R(3,10)$ and $Q(6,4)$:
$m_{RQ}=\frac{4 - 10}{6 - 3}=\frac{-6}{3}=-2$

Step3: Calculate distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $S(2,7)$ and $P(5,1)$:
$SP=\sqrt{(5 - 2)^2+(1 - 7)^2}=\sqrt{3^2+( - 6)^2}=\sqrt{9 + 36}=\sqrt{45}$
For $R(3,10)$ and $Q(6,4)$:
$RQ=\sqrt{(6 - 3)^2+(4 - 10)^2}=\sqrt{3^2+( - 6)^2}=\sqrt{9+36}=\sqrt{45}$
Since $SP$ and $RQ$ are parallel (same slope) and congruent ($SP = RQ=\sqrt{45}$), $PQRS$ is a parallelogram.

Answer:

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