Answer :
To determine if the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other, let’s analyze the given information step by step.
### Step 1: Verify Perpendicularity
1. Slope of Diagonals:
We are given:
- 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 confirm that these diagonals are perpendicular, their slopes should be negative reciprocals of each other. Specifically, the product of the slopes of perpendicular lines is [tex]\(-1\)[/tex].
[tex]\[ 7 \times \left(-\frac{1}{7}\right) = -1 \][/tex]
Since this condition is satisfied, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular.
### Step 2: Verify Bisecting Each Other
2. Midpoint of Diagonals:
We are given that the midpoint of both diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
To confirm that the diagonals bisect each other, they must have the same midpoint.
Given:
- Midpoint of [tex]\(\overline{RP}\)[/tex] is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
- Midpoint of [tex]\(\overline{SQ}\)[/tex] is also [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
Since both diagonals share the same midpoint, this confirms that they bisect each other.
### Conclusion
Since the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular (their slopes are negative reciprocals) and they bisect each other (same midpoint), we can conclude that:
- The diagonals of square [tex]\( PQRS \)[/tex] are both perpendicular and bisectors of each other.
Therefore, the statement that proves the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other is verified by:
[tex]\[ \textrm{The slopes of the diagonals are negative reciprocals, and both diagonals have the same midpoint.} \][/tex]
Thus, the correct conclusion is that the diagonals of square [tex]\( PQRS \)[/tex] are indeed perpendicular bisectors of each other.
### Step 1: Verify Perpendicularity
1. Slope of Diagonals:
We are given:
- 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 confirm that these diagonals are perpendicular, their slopes should be negative reciprocals of each other. Specifically, the product of the slopes of perpendicular lines is [tex]\(-1\)[/tex].
[tex]\[ 7 \times \left(-\frac{1}{7}\right) = -1 \][/tex]
Since this condition is satisfied, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular.
### Step 2: Verify Bisecting Each Other
2. Midpoint of Diagonals:
We are given that the midpoint of both diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
To confirm that the diagonals bisect each other, they must have the same midpoint.
Given:
- Midpoint of [tex]\(\overline{RP}\)[/tex] is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
- Midpoint of [tex]\(\overline{SQ}\)[/tex] is also [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
Since both diagonals share the same midpoint, this confirms that they bisect each other.
### Conclusion
Since the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular (their slopes are negative reciprocals) and they bisect each other (same midpoint), we can conclude that:
- The diagonals of square [tex]\( PQRS \)[/tex] are both perpendicular and bisectors of each other.
Therefore, the statement that proves the diagonals of square [tex]\( PQRS \)[/tex] are perpendicular bisectors of each other is verified by:
[tex]\[ \textrm{The slopes of the diagonals are negative reciprocals, and both diagonals have the same midpoint.} \][/tex]
Thus, the correct conclusion is that the diagonals of square [tex]\( PQRS \)[/tex] are indeed perpendicular bisectors of each other.
