Atticus30idn
Answered

IDNLearn.com is designed to help you find the answers you need quickly and easily. Find reliable solutions to your questions quickly and easily with help from our experienced experts.

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 :

GinnyAnsweridn
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.
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.