Question

which statement proves that the diagonals of square pqrs are perpendicular bisectors of each other?
the length of \\( \overline { sp } \\), \\( \overline { pq } \\), \\( \overline { rq } \\), and \\( \overline { sr } \\) are each 5.
the slope of \\( \overline { sp } \\) and \\( \overline { rq } \\) is \\( - \frac { 4 } { 3 } \\) and the slope of \\( \overline { sr } \\)
and \\( \overline { pq } \\) is \\( \frac { 3 } { 4 } \\).
the length of \\( \overline { sq } \\) and \\( \overline { rp } \\) are both \\( \sqrt { 50 } \\).
the midpoint of both diagonals is \\( \left( 4 \frac { 1 } { 2 }, 5 \frac { 1 } { 2 } \
ight) \\), the slope
of \\( \overline { rp } \\) is 7, and the slope of \\( \overline { sq } \\) is \\( - \frac { 1 } { 7 } \\).

Explanation:

Step1: Recall the properties of perpendicular bisectors

For two lines to be perpendicular bisectors of each other, two conditions must be met:

1. They must bisect each other (i.e., have the same mid - point).
2. The product of their slopes must be \(- 1\) (if the slopes are \(m_1\) and \(m_2\), then \(m_1\times m_2=-1\)).

Step2: Analyze each option

• Option 1:

The length of \(\overline{SP},\overline{PQ},\overline{RQ},\) and \(\overline{SR}\) being \(5\) only shows that the sides of the quadrilateral are equal. It does not give any information about the diagonals (bisecting or perpendicularity).

• Option 2:

The slopes of \(\overline{SP}\) and \(\overline{RQ}\) and \(\overline{SR}\) and \(\overline{PQ}\) are about the sides of the square, not the diagonals.

• Option 3:

The length of \(\overline{SQ}\) and \(\overline{RP}\) being \(\sqrt{50}\) only shows that the diagonals are equal in length. It does not show that they bisect each other or are perpendicular.

• Option 4:
• Mid - point check:

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For diagonal \(\overline{SQ}\) with \(S(1,6)\) and \(Q(8,5)\), the mid - point \(M_{SQ}=(\frac{1 + 8}{2},\frac{6+5}{2})=(4.5,5.5)=(4\frac{1}{2},5\frac{1}{2})\).
For diagonal \(\overline{RP}\) with \(R(5,9)\) and \(P(4,2)\), the mid - point \(M_{RP}=(\frac{5 + 4}{2},\frac{9+2}{2})=(4.5,5.5)=(4\frac{1}{2},5\frac{1}{2})\). So, the diagonals bisect each other.

• Slope check:

The slope formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
The slope of \(\overline{RP}\) with \(R(5,9)\) and \(P(4,2)\) is \(m_{RP}=\frac{9 - 2}{5 - 4}=\frac{7}{1}=7\).
The slope of \(\overline{SQ}\) with \(S(1,6)\) and \(Q(8,5)\) is \(m_{SQ}=\frac{5 - 6}{8 - 1}=\frac{-1}{7}=-\frac{1}{7}\).
Since \(m_{RP}\times m_{SQ}=7\times(-\frac{1}{7})=-1\), the diagonals are perpendicular.

Answer:

The mid - point of both diagonals is \((4\frac{1}{2},5\frac{1}{2})\), the slope of \(\overline{RP}\) is \(7\), and the slope of \(\overline{SQ}\) is \(-\frac{1}{7}\).