To show that the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other, we need to verify two things:
1. The Diagonals Bisect Each Other: This means that the diagonals intersect at the same midpoint.
2. The Diagonals are Perpendicular: This means that the product of their slopes is [tex]\(-1\)[/tex].
Let's start with the information given:
- The lengths of [tex]\(\overline{SP}, \overline{PQ}, \overline{RQ},\)[/tex] and [tex]\(\overline{SR}\)[/tex] are each 5.
- The slopes of [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] are both [tex]\(-\frac{4}{3}\)[/tex].
- The slopes of [tex]\(\overline{SR}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] are both [tex]\(\frac{3}{4}\)[/tex].
- The lengths of [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are both [tex]\(\sqrt{50} \approx 7.071\)[/tex].
- The midpoint of both diagonals is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], which evaluates to [tex]\((4.5, 5.5)\)[/tex].
- The slopes of [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are [tex]\(7\)[/tex] and [tex]\(-\frac{1}{7}\)[/tex], respectively.
### Verifying the Diagonals Bisect Each Other:
The midpoint of both diagonals (given) is [tex]\((4.5, 5.5)\)[/tex]. This indicates that:
- The midpoint of [tex]\(\overline{SQ}\)[/tex] is [tex]\((4.5, 5.5)\)[/tex].
- The midpoint of [tex]\(\overline{RP}\)[/tex] is also [tex]\((4.5, 5.5)\)[/tex].
Since both diagonals share the same midpoint, [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] bisect each other.
### Verifying the Diagonals are Perpendicular:
For the diagonals to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex].
- The slope of [tex]\(\overline{RP}\)[/tex] is [tex]\(7\)[/tex].
- The slope of [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].
Let's multiply the slopes:
[tex]\[ \text{slope of } \overline{RP} \times \text{slope of } \overline{SQ} = 7 \times -\frac{1}{7} = -1 \][/tex]
Since the product of their slopes is indeed [tex]\(-1\)[/tex], the diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are perpendicular.
### Conclusion:
Given that:
- The diagonals share the same midpoint, [tex]\((4.5, 5.5)\)[/tex], proving that they bisect each other.
- The product of the slopes of the diagonals is [tex]\(-1\)[/tex], proving that they are perpendicular.
Hence, we can conclude that the diagonals of square [tex]\(PQRS\)[/tex] are perpendicular bisectors of each other.
