To prove that the diagonals of square PQRS are perpendicular bisectors of each other, we need to demonstrate two things:
1. The diagonals bisect each other.
2. The diagonals are perpendicular to each other.
Given:
- 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].
- The midpoint of both diagonals is [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex].
Step 1: Proving the diagonals are perpendicular
To check if the diagonals are perpendicular, we need to determine if the product of their slopes is [tex]\(-1\)[/tex]. If two lines are perpendicular, the product of their slopes is exactly [tex]\(-1\)[/tex].
Calculate the product of the slopes:
[tex]\[
\text{Product of the slopes} = 7 \times -\frac{1}{7} = -1
\][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are indeed perpendicular to each other.
Step 2: Proving the diagonals bisect each other
To confirm that the diagonals bisect each other, we need to verify that they share the same midpoint. Given the coordinates:
[tex]\[
\text{Midpoint of both diagonals} = \left(4 \frac{1}{2}, 5 \frac{1}{2}\right)
\][/tex]
Since the midpoint of [tex]\(\overline{RP}\)[/tex] is the same as the midpoint of [tex]\(\overline{SQ}\)[/tex], this confirms that both diagonals bisect each other.
Conclusion:
From the given information, we have:
- The product of the slopes of [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] is [tex]\(-1\)[/tex], proving they are perpendicular.
- The midpoints of the diagonals are identical, confirming that the diagonals bisect each other.
Therefore, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] of square PQRS are perpendicular bisectors of each other.
