Answer :
Let's go through the steps to determine if the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other:
1. Length of the Diagonals:
The problem states that the lengths of the diagonals [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] are both [tex]\( \sqrt{50} \)[/tex].
Since the lengths are equal, this indicates that these diagonals bisect each other, as diagonals of a square are always equal. So, we confirm:
- Lengths of diagonals: Equal (yes).
2. Midpoint of the Diagonals:
The midpoint of both diagonals is given as [tex]\( \left(4 \frac{1}{2}, 5 \frac{1}{2}\right) \)[/tex].
For diagonals to bisect each other, they must share the same midpoint. Here, it's confirmed that the midpoints of both [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] are indeed the same. So, this supports these diagonals being perpendicular bisectors:
- Midpoint of diagonals: Equal (yes).
3. Slope of the Diagonals:
The slope of [tex]\( \overline{RP} \)[/tex] is 7, and the slope of [tex]\( \overline{SQ} \)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].
For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. By multiplying the slopes:
[tex]\[ 7 \times \left(-\frac{1}{7}\right) = -1. \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the diagonals are perpendicular to each other.
- Product of slopes: [tex]\(-1\)[/tex] (yes).
Since all criteria have been satisfied—equal lengths of diagonals, equal midpoints, and product of slopes being [tex]\(-1\)[/tex]—we can conclude that the diagonals [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] of square [tex]\( PQRS \)[/tex] are indeed perpendicular bisectors of each other.
Conclusion: The statement that proves the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other is:
"The lengths of [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] are both [tex]\( \sqrt{50} \)[/tex], 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]."
1. Length of the Diagonals:
The problem states that the lengths of the diagonals [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] are both [tex]\( \sqrt{50} \)[/tex].
Since the lengths are equal, this indicates that these diagonals bisect each other, as diagonals of a square are always equal. So, we confirm:
- Lengths of diagonals: Equal (yes).
2. Midpoint of the Diagonals:
The midpoint of both diagonals is given as [tex]\( \left(4 \frac{1}{2}, 5 \frac{1}{2}\right) \)[/tex].
For diagonals to bisect each other, they must share the same midpoint. Here, it's confirmed that the midpoints of both [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] are indeed the same. So, this supports these diagonals being perpendicular bisectors:
- Midpoint of diagonals: Equal (yes).
3. Slope of the Diagonals:
The slope of [tex]\( \overline{RP} \)[/tex] is 7, and the slope of [tex]\( \overline{SQ} \)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].
For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. By multiplying the slopes:
[tex]\[ 7 \times \left(-\frac{1}{7}\right) = -1. \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the diagonals are perpendicular to each other.
- Product of slopes: [tex]\(-1\)[/tex] (yes).
Since all criteria have been satisfied—equal lengths of diagonals, equal midpoints, and product of slopes being [tex]\(-1\)[/tex]—we can conclude that the diagonals [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] of square [tex]\( PQRS \)[/tex] are indeed perpendicular bisectors of each other.
Conclusion: The statement that proves the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other is:
"The lengths of [tex]\( \overline{SQ} \)[/tex] and [tex]\( \overline{RP} \)[/tex] are both [tex]\( \sqrt{50} \)[/tex], 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]."
