The first step in using slope is to determine the change in y and the change in x. We can do that by subtracting the adjacent coordinates. If we do that in the same order, B-A, C-B, D-C, A-D, then we should expect to see alternate differences have alternate signs, but the same values.
Since adjacent edges are supposed to be perpendicular, we should see their slopes be negative reciprocals.
B-A = (1, 8) - (-5, 5) = (6, 3) . . . . slope = 3/6 = 1/2
C-B = (4, 2) - (1, 8) = (3, -6) . . . . slope = -6/3 = -2
D-C = (-2, -2) - (4, 2) = (-6, -4) . . . . slope = -4/-6 = 2/3 . . . . . not a rectangle
(2/3 is not the same as 1/2, nor is it the negative reciprocal of -2. Segment CD is not parallel to AB, nor is it perpendicular to BC.)
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You may be able to reach this conclusion faster by summing the coordinates of opposite vertices: A+C = (-5, 5) + (4, 2) = (-1, 7); B+D = (1, 8) + (-2, -2) = (-1, 6). In a rectangle, these sums are equal, because both sums are twice the coordinates of the midpoint of the corresponding diagonal. The diagonals of a rectangle have the same midpoint. These different sums mean the figure is not a rectangle. (However, the problem specified the method, so this simpler method can only be used as a check.)
