Which of the following could be the coordinates of the fourth vertex?

Question

The whole question is this:
Three of the four vertices of a parallelogram are located at (1,1), (7,1), and (3,5). Which of the following could be the coordinates of the fourth vertex?
I. (-3,5)
II. (5,5)
III. (9,5)

A. I only
B. III only
C. I and II only
D. I and III only
E. II and III only

Please help. Thank you!

Answer 1

This problem is so cool to see with color computer graphics, bur graphing the points help a whole lot, so do that.
 
With three points (essentially a triangle), you may add any of three other points that will make a parallelogram (plot this;  the new point completes the parallelogram by adding a new triangle to each of the three sides).
 
Since the problem said, "parallelogram," we know that the differences of the coordinates of two points is the same as the differences of the coordinates of the two other points (again, check this on your graph!).
 
So, if you have (1,1) and (7,1), this is a horizontal line.  The horizontal line on the other side of the parallelogram goes from (3,5) in the same direction and for the same length.  It ends at (9,5)  [note change of x is 6, change of y is 0).
 
If you have (3,5) and (7,1), the change in x is 4 and the change in y is -4, so the line goes from (1,1) to (5,-3).
 
And, if you have (7,1) and (3,5), the change in x is -4 and the change in y is 4, so the line goes from (1,1) to (-3,5). There are three possible new vertices.

[note: by now you should get the "picture" and be able to choose the correct answer.]

[another note: notice that I started each of the three segments at a different point and did not use any line twice]