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

A. The lengths 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]$\overline{SF}$[/tex] and [tex]$\overline{PQ}$[/tex] is [tex]$\frac{3}{4}$[/tex].

C. The lengths 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].



Answer :

GinnyAnswer
To prove that the diagonals of square PQRS are perpendicular bisectors of each other, we need to show that the diagonals intersect at right angles. In other words, if the slopes of the diagonals are negative reciprocals of each other, it will prove that they are perpendicular.

Given the slopes:
- The slope of line [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex].
- The slope of line [tex]\(\overline{SF}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].

To determine if the diagonals are perpendicular, we consider the product of the slopes. For two lines with slopes [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] to be perpendicular, the product [tex]\(m_1 \times m_2\)[/tex] should be [tex]\(-1\)[/tex].

Let's check this:

For the slopes [tex]\(-\frac{4}{3}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ \left(-\frac{4}{3}\right) \times \left(\frac{3}{4}\right) = -1 \][/tex]

Since the product of the slopes is [tex]\(-1\)[/tex], this confirms that the diagonals are indeed perpendicular.

Hence, the statement that proves the diagonals are perpendicular is:
"The slope of [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex] and the slope of [tex]\(\overline{SF}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex]." This confirms that the diagonals are perpendicular bisectors of each other.

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