Short Answer 
The proof shows that the diagonals of the square are perpendicular bisectors by calculating their slopes and midpoints. The slopes of the diagonals PS and QR are negative reciprocals, and both share the midpoint (2, 2), confirming they are perpendicular to each other and bisect each other.
Step 1: Calculate the Slope of the Diagonals
To prove that the diagonals of the square are perpendicular bisectors, first, determine the slopes of the diagonals. The square has vertices P(0, 4), Q(4, 4), R(0, 0), and S(4, 0). You can calculate the slopes as follows:
- The slope of diagonal PS is -1.
- The slope of diagonal QR is 1.
Step 2: Determine the Midpoint of the Diagonals
Next, find the midpoints of both diagonals. Midpoints are essential to show that the diagonals bisect each other. The calculations yield:
- The midpoint of diagonal PS is at (2, 2).
- The midpoint of diagonal QR is also at (2, 2).
Step 3: Conclude the Proof
Finally, combine the results from the slopes and midpoints to establish that the diagonals are perpendicular bisectors. Since the slopes of PS and QR are negative reciprocals of each other and they share the same midpoint, this confirms that:
- The diagonals are perpendicular to each other.
- The diagonals bisect each other at the point (2, 2).
- Thus, they are indeed perpendicular bisectors.