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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 determine if the diagonals of square PQRS (specifically diagonals RP and SQ) are perpendicular bisectors of each other, we need to check the product of the slopes of these diagonals. If the product of their slopes is -1, then the diagonals are perpendicular.

Here's a step-by-step explanation:

1. Identify the slopes of the diagonals of the square:
- The slope of diagonal RP is given as \(7\).
- The slope of diagonal SQ is given as \(-\frac{1}{7}\).

2. Calculate the product of these slopes:
- Multiply the slope of RP by the slope of SQ:
[tex]\[ \text{Product of slopes} = 7 \times \left( -\frac{1}{7} \right) \][/tex]
The product is:
[tex]\[ 7 \times -\frac{1}{7} = -1 \][/tex]

3. Determine if the diagonals are perpendicular:
- In geometry, two lines are perpendicular if the product of their slopes is \(-1\).
- Since the product of the slopes \( = -1\), this confirms that diagonals RP and SQ are indeed perpendicular.

4. Conclusion:
- Thus, the statement that proves diagonals of square PQRS are perpendicular bisectors of each other is correct based on the given slopes of the diagonals.

Therefore, the diagonals RP and SQ are perpendicular bisectors of each other because the product of their slopes is [tex]\(-1\)[/tex].
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