Question

On a coordinate plane, square P Q R S is shown. It has points (0, 4), (4, 4), (0, 0), and (4, 0).
Prove the diagonals of the square with vertices P(0, 4), Q(4, 4), R(0, 0), and S(4, 0) are perpendicular bisectors of each other.
Step 1: Calculate the slope of the diagonals.

The slope of diagonal PS is
.

The slope of diagonal QR is
.

Step 2: Calculate the midpoint of the diagonals.

The midpoint of PS is
.

The midpoint of QR is
.

The diagonals of the square are perpendicular bisectors because the diagonals are
.

Answer 1

Step 1:
Slope of PS = (0 − 4)/(4 − 0) = −4/4 = −1.
Slope of QR = (0 − 4)/(0 − 4) = −4/−4 = 1.

Step 2:
Midpoint of PS = ((0+4)/2, (4+0)/2) = (2, 2).
Midpoint of QR = ((4+0)/2, (4+0)/2) = (2, 2).

The diagonals are perpendicular bisectors because their slopes are negative reciprocals (−1 and 1), so they are perpendicular, and they share the same midpoint (2,2), so each bisects the other.