The vertices of square pqrs are p(-4,0) q(4,3) r(7,-5) s(-1,-8). show that the diagonals of square pqrs are congruent perpendicular bisectors of each other.

To show that the diagonals of square PQRS are congruent is to show that they are equal.
|PR| = sqrt((7 - (-4))^2 + (-5 - 0)^2) = sqrt(11^2 + (-5)^2) = sqrt(146)
|QS| = sqrt((-1 - 4)^2 + (-8 - 3)^2) = sqrt((-5)^2 + (-11)^2) = sqrt(146)

To show that they are perpendicular, then the product of their slopes is -1.
slope of |PR| = (-5 - 0)/(7 - (-4)) = -5/11
slope of |QS| = (-8 - 3)/(-1 - 4) = -11/-5 = 11/5
The product of their slopes = -5/11 * 11/5 = -1

To show that they are bisectors is to show that their midpoints are at the same point.
Midpoint of |PR| = ((7 - 4))/2, (-5 + 0)/2) = (3/2, -5/2)
Midpoint of |QS| = ((-1 + 4)/2, (3 - 8)/2) = (3/2, -5/2)