Question

which statement proves that pqrs is a parallelogram?
the slopes of \\( \overline { s p } \\) and \\( \overline { r q } \\) are both -2 and \\( s p = r q \\)
\\( = \sqrt { 45 } \\).
the slopes of \\( \overline { r s } \\) and \\( \overline { q p } \\) are both 3 and \\( s p = r q = \\)
\\( \sqrt { 45 } \\).
the midpoint of \\( \overline { r p } \\) is \\( \left( 4,5 \frac { 1 } { 2 } \
ight) \\) and the slope of \\( \overline { r p } \\) is
\\( - \frac { 9 } { 2 } \\).
the midpoint of \\( \overline { s q } \\) is \\( \left( 4,5 \frac { 1 } { 2 } \
ight) \\) and \\( s q = 5 \\).

Explanation:

Step1: Recall the properties of a parallelogram

One of the properties of a parallelogram is that if one pair of opposite sides is both parallel (same slope) and congruent (same length), then the quadrilateral is a parallelogram.

Step2: Analyze each option

• Option 1:

Slopes of \(SP\) and \(RQ\) are considered. But \(SP\) and \(RQ\) are not opposite sides of the quadrilateral \(PQRS\).

• Option 2:

For two - points \((x_1,y_1)\) and \((x_2,y_2)\), the slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \(R(3,10)\) and \(S(2,7)\), slope of \(RS\) is \(m_{RS}=\frac{7 - 10}{2 - 3}=\frac{- 3}{-1}=3\).
For \(Q(6,4)\) and \(P(5,1)\), slope of \(QP\) is \(m_{QP}=\frac{1 - 4}{5 - 6}=\frac{-3}{-1}=3\).
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 \(P(5,1)\), \(SP=\sqrt{(5 - 2)^2+(1 - 7)^2}=\sqrt{9 + 36}=\sqrt{45}\).
For \(R(3,10)\) and \(Q(6,4)\), \(RQ=\sqrt{(6 - 3)^2+(4 - 10)^2}=\sqrt{9+36}=\sqrt{45}\).
Since \(RS\parallel QP\) (same slope \(m = 3\)) and \(SP = RQ=\sqrt{45}\), by the property of a parallelogram (one pair of opposite sides is parallel and congruent), \(PQRS\) is a parallelogram.

• Option 3:

Information about the mid - point and slope of \(RP\) does not help in proving that \(PQRS\) is a parallelogram using the standard parallelogram properties (e.g., opposite sides parallel and congruent, diagonals bisecting each other etc.). The given data about \(RP\) is not relevant for the parallelogram criteria.

• Option 4:

Information about the mid - point and length of \(SQ\) does not help in proving that \(PQRS\) is a parallelogram using the standard parallelogram properties (e.g., opposite sides parallel and congruent, diagonals bisecting each other etc.). The given data about \(SQ\) is not relevant for the parallelogram criteria.

Answer:

The slopes of \(\overline{RS}\) and \(\overline{QP}\) are both \(3\) and \(SP = RQ=\sqrt{45}\).