Question

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.

Answer 1

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)

Answer 2

-1

1

(2,2)

(2,2)

Perpendicular and share same midpoint