Question

Three of the vertices of a parallelogram are (2, 1) (4, 7), and (6, 5). What are the three possible coordinates of the fourth vertex? (0, 3) (-1, 4) (3, 3) (4, -1) (4, 4) (8, 11)

Answer 1

You can do this without drawing even. Let's say the coordinates are
A, B and C, respectively, and we're looking for D.

If the other vertex is opposite A(2,1), it will be at B+C-A = (8, 11)
If the other vertex is opposite B, it will be at A+C-B = (4, -1)
If the other vertex is opposite C, it will be at A+B-C = (0, 3)

Explanation: think of one of the vertexes as a corner of the parallelogram and you want to travel to the opposite corner. That means you must add both vectors that represent the sides. So from A to D, you'd have to add (B-A) and (C-A). That gives us A+(B-A)+(C-A) simplifies to B+C-A.