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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 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.
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