To prove that the diagonals of square \(PQRS\) are perpendicular bisectors of each other, we need to show two main properties:
1. The diagonals bisect each other, which means they intersect at their midpoints.
2. The diagonals are perpendicular to each other.
Let's analyze each statement to see which one provides these proofs:
Statement (a): The length of \(\overline{SP}\), \(\overline{PQ}\), \(\overline{RQ}\), and \(\overline{SR}\) are each 5.
- This statement tells us the sides of the square are all equal, but it does not provide information about the properties of the diagonals (that they bisect each other or are perpendicular).
Statement (b): The slope of \(\overline{SP}\) and \(\overline{RQ}\) is \( -\frac{4}{3} \), and the slope of \(\overline{SR}\) and \(\overline{PQ}\) is \( \frac{3}{4} \).
- This statement provides the slopes of the sides of the square. While it might help in establishing that the sides are perpendicular to each other, this does not help in proving the properties of the diagonals.
Statement (c): The length of \(\overline{SQ}\) and \(\overline{RP}\) are both \(\sqrt{50}\).
- This statement tells us the lengths of the diagonals are equal, which is expected in a square. However, it does not confirm that the diagonals bisect each other or that they are perpendicular.
Statement (d): The midpoint of both diagonals is \(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\), the slope of \(\overline{RP}\) is 7, and the slope of \(\overline{SQ}\) is \( -\frac{1}{7} \).
- This statement provides crucial information:
- Both diagonals intersect at the midpoint \(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\).
- The slopes of the diagonals are 7 and \( -\frac{1}{7} \).
The fact that the product of the slopes of the diagonals (\(7 \cdot -\frac{1}{7}\)) is \(-1\) confirms that the diagonals are perpendicular. Additionally, both diagonals sharing the same midpoint confirms that they bisect each other. Hence, statement (d) proves that the diagonals of square \(PQRS\) are perpendicular bisectors of each other.
Therefore, the correct statement is
(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].
