Which statement proves that the diagonals of square PQRS are perpendicular bisectors of each other?

A. The length of [tex]$\overline{SP}$[/tex], [tex]$\overline{PQ}$[/tex], [tex]$\overline{RQ}$[/tex], and [tex]$\overline{SR}$[/tex] are each 5.

B. The slope of [tex]$\overline{SP}$[/tex] and [tex]$\overline{RQ}$[/tex] is [tex]$-\frac{4}{3}$[/tex], and the slope of [tex]$\overline{PQ}$[/tex] and [tex]$\overline{SR}$[/tex] is [tex]$\frac{3}{4}$[/tex].

C. The length of [tex]$\overline{SQ}$[/tex] and [tex]$\overline{RP}$[/tex] are both [tex]$\sqrt{50}$[/tex].

D. The midpoint of both diagonals is [tex]$\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)$[/tex], the slope of [tex]$\overline{RP}$[/tex] is 7, and the slope of [tex]$\overline{SQ}$[/tex] is [tex]$-\frac{1}{7}$[/tex].



Answer :

To prove that the diagonals of square PQRS are perpendicular bisectors of each other, we need to demonstrate two things:
1. The diagonals bisect each other.
2. The diagonals are perpendicular to each other.

Given:
- The slope of diagonal [tex]\(\overline{RP}\)[/tex] is [tex]\(7\)[/tex].
- The slope of diagonal [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].
- The midpoint of both diagonals is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].

Step 1: Proving the diagonals are perpendicular

To check if the diagonals are perpendicular, we need to determine if the product of their slopes is [tex]\(-1\)[/tex]. If two lines are perpendicular, the product of their slopes is exactly [tex]\(-1\)[/tex].

Calculate the product of the slopes:
[tex]\[ \text{Product of the slopes} = 7 \times -\frac{1}{7} = -1 \][/tex]

Since the product of the slopes is [tex]\(-1\)[/tex], the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are indeed perpendicular to each other.

Step 2: Proving the diagonals bisect each other

To confirm that the diagonals bisect each other, we need to verify that they share the same midpoint. Given the coordinates:
[tex]\[ \text{Midpoint of both diagonals} = \left(4 \frac{1}{2}, 5 \frac{1}{2}\right) \][/tex]

Since the midpoint of [tex]\(\overline{RP}\)[/tex] is the same as the midpoint of [tex]\(\overline{SQ}\)[/tex], this confirms that both diagonals bisect each other.

Conclusion:

From the given information, we have:
- The product of the slopes of [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] is [tex]\(-1\)[/tex], proving they are perpendicular.
- The midpoints of the diagonals are identical, confirming that the diagonals bisect each other.

Therefore, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] of square PQRS are perpendicular bisectors of each other.