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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 prove that the diagonals of square PQRS are perpendicular bisectors of each other, we must demonstrate the following:

1. The diagonals bisect each other.
2. The diagonals are equal in length.
3. The diagonals are perpendicular.

From the given information:

1. Midpoint Analysis:
- The midpoint of both diagonals is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
- This common midpoint for both diagonals indicates that the diagonals bisect each other.

2. Length of Diagonals:
- The length of [tex]\(\overline{SQ}\)[/tex] is [tex]\(\sqrt{50}\)[/tex].
- The length of [tex]\(\overline{RP}\)[/tex] is also [tex]\(\sqrt{50}\)[/tex].
- Since both diagonals have the same length, they are equal.

3. Slopes of the Diagonals:
- The slope of [tex]\(\overline{RP}\)[/tex] is 7.
- The slope of [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].
- The product of the slopes of two perpendicular lines is -1: [tex]\(7 \times \left(-\frac{1}{7}\right) = -1\)[/tex].
- This confirms that the diagonals are perpendicular to each other.

Therefore, based on midpoint analysis, equal lengths, and perpendicularity, the diagonals of square PQRS bisect each other, are equal in length, and are perpendicular. This conclusively proves that the diagonals are perpendicular bisectors of each other.
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