The opposite sides of the quadrilateral should be parallel and the diagonals should be equal. [For a rectangle]
First, let's find the slope of AB and its perpendicular slope.
Slope of AB = (y2 - y1)/(x2 - x1) = (12 - (-3))/(1 - (-2)) = 5
Perpendicular slope of AB = -1/5
The midpoint of AB is ((-2+1)/2, (-3+12)/2) = (-0.5, 4.5)
Let D be the point on line segment AC such that AD is perpendicular to AC. Since AB is parallel to DC, the slope of DC is also 5.
Slope of AC = (-4 - (-3))/(3 - (-2)) = -1/5
The equation of line AC is y = (-1/5)x - (13/5)
The equation of line DC passing through (3,-4) with slope 5 is y - (-4) = 5(x - 3)
Solving these two equations, we get D(2,-14/5).
Therefore, the coordinates of point D are (2, -14/5).
We can plot the points on the coordinate axes and verify that A, B, C, and D indeed form a rectangle.