No diagram has been provided, so the following solution is based on guessing what the diagram looks like.
AC and AR must be perpendicular to form a rectangle. Graphically, we can plot A(0,2) and C(-4,0) and draw a line through A and C and a circle with centre (0,2) and radius 5 (length of AR): x2+(y-2)2=25 (green on graph) is the equation of the circle. The equation of the line passing through A and C is y=x/2+2 (red) and the equation of AR is y=-2x+2 (blue).
The coordinates of R can be found by substituting for y in the equation of the circle:
x2+(-2x+2-2)2=25, x2+4x2=25, so 5x2=25, x2=5, making x=±√5 (orange lines), and y=∓2√5+2.
Diagram below:
So two rectangles are possible with 3 of the vertices: A(0,2), C(-4,0), R1(√5,-2√5+2) or R2(-√5,2√5+2).
I'm guessing that c is a side opposite to right angle C, so c is another name for AR. If M is the midpoint, then clearly there will be two midpoints depending on the location of R. M will be the average of the coordinates of A and R.
M=½(±√5,∓2√5+4)=(±√5/2,∓√5+2), that is, M1(√5/2,-√5+2) or M2(-√5/2,√5+2).
As you can see I've had to make various assumptions (which may or may not be correct) because the question is unclear and no diagram has been provided.