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

A. The length of [tex] \overline{SP}, \overline{PQ}, \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{SR} [/tex] and [tex] \overline{PQ} [/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, let's analyze the given information step by step.

### Step-by-Step Solution

1. Lengths of the Sides:
- We know the lengths of all sides of the square PQRS are equal to 5 units. Hence,
[tex]\[ \overline{SP} = \overline{PQ} = \overline{RQ} = \overline{SR} = 5 \][/tex]

2. Slopes of the Sides:
- The slope of [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] is:
[tex]\[ -\frac{4}{3} \][/tex]
- The slope of [tex]\(\overline{SR}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] is:
[tex]\[ \frac{3}{4} \][/tex]

3. Lengths of the Diagonals:
- The lengths of both diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are:
[tex]\[ \sqrt{50} \][/tex]

4. Midpoint of the Diagonals:
- Both diagonals have a common midpoint given by:
[tex]\[ \left( 4 \frac{1}{2}, 5 \frac{1}{2} \right) \][/tex]

5. Slopes of the Diagonals:
- The slope of the diagonal [tex]\(\overline{RP}\)[/tex] is:
[tex]\[ 7 \][/tex]
- The slope of the diagonal [tex]\(\overline{SQ}\)[/tex] is:
[tex]\[ -\frac{1}{7} \][/tex]

6. Proof of Perpendicularity:
- For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. Let's check the product of the slopes of the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex]:
[tex]\[ \text{slope}_{\overline{RP}} \times \text{slope}_{\overline{SQ}} = 7 \times -\frac{1}{7} = -1 \][/tex]
- Therefore, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular.

7. Proof of Bisection:
- Since both diagonals have the same lengths ([tex]\(\sqrt{50}\)[/tex]) and share a common midpoint [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], it means they bisect each other at this point. This is because the midpoint is equidistant from the endpoints of both diagonals.

### Conclusion
- The diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular because their slopes multiply to [tex]\(-1\)[/tex].
- The diagonals bisect each other at [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], as they share a common midpoint.
- Therefore, the diagonals of square PQRS are perpendicular bisectors of each other, thus confirming the properties of the diagonals in a square.

Hence, all the conditions are satisfied, proving that the diagonals of square PQRS are perpendicular bisectors of each other.