SOLUTION: 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?

Question 821011: 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?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The three given points are in red,and I connected them into a red triangle.
Parallelogram sides parallel to those 3 red sides,form a similar triangle and the vertices of that larger triangle are the possible fourth vertices for the parallelogram.
Point completes the green (and red) parallelogram, which happens to be a rectangle.
Point completes the blue (and red) parallelogram.
Point completes the black (and red) parallelogram.

Drawing is easier than calculating coordinates.
In the parallelogram, one of the 3 given points must be connected to both of the other ones, and the other ones form the diagonal of the parallelogram.
If point (6,5) is connected to (2, 1) and (4, 7),
the line connecting (2, 1) and (4, 7) is a diagonal of the parallelogram.
If

is a parallelogram, then is a diagonal.
To go from to the fourth vertex, ,
we go on the parallel side opposite

the same distance and direction as from to .
Going from A(4,7) to one 4th vertex, is the same as from B(6,5) to C(2,1),
4 units left and 4 down, and we find X(0,3).
Going from A(4,7) to the 4th vertex, is the same as from B(2,1) to C(6,5),
4 units right and 4 up, and we find Y(8,11).
Going from A(2,1) to the 4th vertex, is the same as from B(4,7) to C(6,5),
2 units right and 2 down, and we find Z(4,-1).
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