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

A. The length of [tex]$\overline{SP}, \overline{PQ}, \overline{RQ},$[/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 need to demonstrate two primary properties:

1. Perpendicularity: The diagonals should intersect at a right angle (90 degrees).
2. Bisection: The diagonals should split each other into equal lengths at the point of intersection.

Let's detail each criterion step-by-step:

### Step 1: Calculate the Lengths and Midpoint of the Diagonals

Given data:
- The length of segments [tex]\(\overline{SP}\)[/tex], [tex]\(\overline{PQ}\)[/tex], [tex]\(\overline{RQ}\)[/tex], and [tex]\(\overline{SR}\)[/tex] are each 5. This suggests that PQRS is a square, as all its sides are equal and presumably its angles are right angles.
- The length of both diagonals, [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex], is [tex]\(\sqrt{50}\)[/tex], which is approximately [tex]\(7.071\)[/tex].

### Step 2: Perpendicularity of the Diagonals

To check if the diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are perpendicular, we observe their slopes.

- The slope of diagonal [tex]\(\overline{RP}\)[/tex] is [tex]\(7\)[/tex].
- The slope of diagonal [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex].

Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Calculating the product of the slopes:
[tex]\[ 7 \times \left(-\frac{1}{7}\right) = -1 \][/tex]
Since the product of the slopes is indeed [tex]\(-1\)[/tex], the diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are perpendicular.

### Step 3: Bisection at the Midpoint

For the diagonals to be bisectors, they must intersect and divide each other into two equal parts at their midpoint. The coordinates of the midpoint help us verify this.

- The given midpoint of both diagonals is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].

Since both diagonals share the same midpoint, it means they intersect at this point and each diagonal is split into two equal parts by the other.

### Step 4: Conclusion

The lengths of all sides of the square are equal, the diagonals are both [tex]\(\sqrt{50}\)[/tex], they bisect each other at the midpoint [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], and the slopes of the diagonals multiply to [tex]\(-1\)[/tex], establishing that they are perpendicular.

Thus, the calculations and the given data confirm that the diagonals of square PQRS are perpendicular bisectors of each other.
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