To prove that the diagonals of square PQRS are perpendicular bisectors of each other, we need to verify two main points:
1. The diagonals are perpendicular to each other.
2. The diagonals bisect each other.
### Step 1: Checking Perpendicularity of Diagonals
To determine if two lines are perpendicular, the product of their slopes should be -1.
- Given the slopes of the diagonals [tex]\( \overline{RP} \)[/tex] and [tex]\( \overline{SQ} \)[/tex]:
- Slope of [tex]\( \overline{RP} \)[/tex] is 7
- Slope of [tex]\( \overline{SQ} \)[/tex] is [tex]\( -\frac{1}{7} \)[/tex]
Calculate their product:
[tex]\[ \text{slope\_product} = 7 \times \left(-\frac{1}{7}\right) = -1 \][/tex]
Since the product of the slopes is -1, the diagonals [tex]\( \overline{RP} \)[/tex] and [tex]\( \overline{SQ} \)[/tex] are perpendicular.
### Step 2: Checking Bisecting Property of Diagonals
For the diagonals to bisect each other, they must intersect at their midpoint. The given information tells us:
- The midpoint of both diagonals is [tex]\( \left(4 \frac{1}{2}, 5 \frac{1}{2}\right) \)[/tex].
This midpoint is common to both [tex]\( \overline{RP} \)[/tex] and [tex]\( \overline{SQ} \)[/tex]. Therefore, the diagonals bisect each other at this point.
### Conclusion
1. Perpendicularity: The product of the slopes of diagonals [tex]\( \overline{RP} \)[/tex] and [tex]\( \overline{SQ} \)[/tex] is -1, indicating they are perpendicular.
2. Bisecting at Midpoint: Both diagonals share a common midpoint at [tex]\( \left(4 \frac{1}{2}, 5 \frac{1}{2}\right) \)[/tex], indicating they bisect each other.
Thus, these two points confirm that the diagonals of square PQRS are perpendicular bisectors of each other.
