SOLUTION: three vertices of a quadrilateral are (-1,-1) (1,2) and (5,-1) what are the coordinates of two vertices that will form two different parallelograms?

You take any two of the points, and observe how you get from one to
the other by moving horizontally or vertically so many units. Then
you start at the third point and move the same number of units
horizontally and vertically and you'll end up at a fourth point.

There are three possible fourth points that form a parallelogram with
the other three. 



If we start at (-1,1) and go to (1,2) we notice that we have to move 2 units
right and then move 3 units up.
Now we start at (5,-1) and do the same thing.  That is, we move 2 units right
and then 3 units up. We then end up at (7,2), the point in red, and draw
the parallelogram (in green):



----------------

If we start at (5,-1) and go to (-1,-1) we notice that we have to move 6 units
left (and NO units up).
Now we start at (1,2) and do the same thing.  That is, we move 6 units left
(and NO units up). We then end up at (-5,2), the point in red, and we draw
the parallelogram (in green):



------------------------------

If we start at (1,2) and go to (-1,1) we notice that we have to move 2 units
left and 3 units down.
Now we start at (5,-1) and do the same thing.  That is, we move 2 units left
and 3 units down. We then end up at (3,-4), the point in red, and we draw
the parallelogram (in green):




Just for fun, let's look at all of them on one graph:



Edwin