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Which statement proves that the diagonals of square PQRS are perpendicular bisectors of each other?

A. The length of [tex]$\overline{SP}$[/tex], [tex]$\overline{PQ}$[/tex], [tex]$\overline{RQ}$[/tex], and [tex]$\overline{SR}$[/tex] are each 5.
B. 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].
C. The length of [tex]$\overline{SQ}$[/tex] and [tex]$\overline{RP}$[/tex] are both [tex]$\sqrt{50}$[/tex].
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].

Sagot :

GinnyAnswer
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.
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