Question

which statement proves that pqrs is a parallelogram?
the slopes of (overline{sp}) and (overline{rq}) are both (-2) and (sp = rq=sqrt{45}).
the slopes of (overline{rs}) and (overline{qp}) are both (3) and (sp = rq=sqrt{45}).
the midpoint of (overline{rp}) is (left(4,5\frac{1}{2}
ight)) and the slope of (overline{rp}) is (-\frac{9}{2}).
the midpoint of (overline{sq}) is (left(4,5\frac{1}{2}
ight)) and (sq = 5).

Explanation:

Step1: Recall the properties of a parallelogram

One of the properties of a parallelogram is that one - pair of opposite sides are both 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: Calculate the slope and length for the first option

For \(S(2,7)\) and \(P(5,1)\):
Slope of \(\overline{SP}\): \(m_{SP}=\frac{1 - 7}{5 - 2}=\frac{-6}{3}=-2\)
Length of \(\overline{SP}\): \(d_{SP}=\sqrt{(5 - 2)^2+(1 - 7)^2}=\sqrt{9 + 36}=\sqrt{45}\)
For \(R(3,10)\) and \(Q(6,4)\):
Slope of \(\overline{RQ}\): \(m_{RQ}=\frac{4 - 10}{6 - 3}=\frac{-6}{3}=-2\)
Length of \(\overline{RQ}\): \(d_{RQ}=\sqrt{(6 - 3)^2+(4 - 10)^2}=\sqrt{9+36}=\sqrt{45}\)
Since \(m_{SP}=m_{RQ}=-2\) and \(SP = RQ=\sqrt{45}\), \(\overline{SP}\parallel\overline{RQ}\) and \(\overline{SP}\cong\overline{RQ}\)

Step3: Check other options

• For the second option:

For \(R(3,10)\) and \(S(2,7)\): \(m_{RS}=\frac{7 - 10}{2 - 3}=3\)
For \(Q(6,4)\) and \(P(5,1)\): \(m_{QP}=\frac{1 - 4}{5 - 6}=3\)
But \(SP\) and \(RQ\) are the sides we need to check for the parallelogram property (not \(RS\) and \(QP\))

• For the third option:

Mid - point formula \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\) for \(R(3,10)\) and \(P(5,1)\): \((\frac{3 + 5}{2},\frac{10 + 1}{2})=(4,5.5)\)
Slope of \(\overline{RP}=\frac{1 - 10}{5 - 3}=-\frac{9}{2}\). This only gives information about the diagonal \(\overline{RP}\), not about the sides.

• For the fourth option:

Mid - point of \(S(2,7)\) and \(Q(6,4)\) is \((\frac{2+6}{2},\frac{7 + 4}{2})=(4,5.5)\) and \(SQ=\sqrt{(6 - 2)^2+(4 - 7)^2}=\sqrt{16 + 9}=5\). This gives information about the diagonal \(\overline{SQ}\), not about the sides.

Answer:

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