To determine if the diagonals of square PQRS are perpendicular bisectors of each other, let's analyze the given information step by step:
1. Lengths of Sides:
The lengths of [tex]\(\overline{SP}, \overline{PQ}, \overline{RQ}\)[/tex], and [tex]\(\overline{SR}\)[/tex] are each [tex]\(5\)[/tex], indicating that PQRS is a square since all sides are equal.
2. Slopes of 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 Diagonals:
The lengths of diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are both [tex]\(\sqrt{50}\)[/tex], suggesting these diagonals are equal in length.
4. Midpoint of Diagonals:
The midpoint of both diagonals is given as [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex]. A common midpoint for both diagonals confirms that they bisect each other.
5. Slopes of Diagonals:
- The slope of [tex]\(\overline{RP}\)[/tex] is [tex]\(7\)[/tex].
- The slope of [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].
To determine if the diagonals are perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]:
[tex]\[
\text{Slope of } \overline{RP} \times \text{Slope of } \overline{SQ} = 7 \times \left(-\frac{1}{7}\right) = -1
\][/tex]
This condition is satisfied.
### Conclusion
Based on the analysis, the following points confirm that the diagonals of square PQRS are perpendicular bisectors of each other:
1. The diagonals have a common midpoint [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\)[/tex]\, meaning they bisect each other.
2. The product of the slopes of the diagonals [tex]\((7 \times -\frac{1}{7} = -1)\)[/tex] confirms that the diagonals are perpendicular.
Therefore, it is proven that the diagonals of square PQRS are perpendicular bisectors of each other.
