Total no. of points in this area = 20

Point A can now take up any of these points.

Rajata's observation provides an interesting/differnet way to tackle this question.
We can reduce our calculations by recognizing that a rectangle in this question is defined by 2 points (rather than 4). All we need are the coordinates of 2 vertices that are diagonal from each other.
For example, the vertices at (2,0) and (5,3) can define only one rectangle.

So, this counting problem is reduced to selecting 2 points in the given region.
The first point can be selected in 20 ways.
Once we have selected the first point, we need to select another point which is diagonal to this point. If this is the case, our second point cannot have the same x- or y-coordinate as the first point. This means that, when selecting the second point, we have only 12 points from which to choose.
So, the total number of ways to select 2 points from the region is 20x12=120 ways.
BUT, we need to divide this by 2 since we have counted each pair of coordinates twice (e.g., selecting points (2,0 and 5,3) is the same as selecting points (5,3) and (2,0).