To prove that the diagonals of square PQRS are perpendicular bisectors of each other, we need to show that:
1. The diagonals bisect each other.
2. The diagonals are perpendicular.
Given the information:
- The length of [tex]\(\overline{SP}\)[/tex], [tex]\(\overline{PQ}\)[/tex], [tex]\(\overline{RQ}\)[/tex], and [tex]\(\overline{SR}\)[/tex] are each 5. This confirms that PQRS is a square.
- 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]. These properties are consistent with the sides of a square having perpendicular slopes.
- The length of [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are both [tex]\(\sqrt{50}\)[/tex]. Since the diagonals of a square are equal, this verifies that [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are indeed the diagonals, and [tex]\(\sqrt{50}\)[/tex] is the correct length for the diagonals of a square with side length 5.
To address the crucial points for perpendicular bisectors:
- The midpoint of both diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex]. This confirms that the diagonals have the same midpoint, meaning they bisect each other.
- The slope of [tex]\(\overline{RP}\)[/tex] is 7, and the slope of [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex]. The product of the slopes of two perpendicular lines is [tex]\(-1\)[/tex], which is indeed the case here, as [tex]\(7 \times -\frac{1}{7} = -1\)[/tex].
Therefore:
1. Since both diagonals have the same midpoint, they bisect each other.
2. Since the slopes of [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] multiply to [tex]\(-1\)[/tex], they are perpendicular.
Hence, it is proven that the diagonals of square PQRS are perpendicular bisectors of each other.
