Question

The coordinates of three vertices of a certain parallelogram are (5,2), 2,6) and (10,9). Give the coordinates of a fourth vertex. Then give the coordinates of another point that could be the fourth vertex.

Answer 1

Plot the points. Clearly (2,6) and (5,2) are the lower left and lower right vertices.
So, (10,9) could be either upper left or upper right.

(2,6) is above and to the left of (5,2)
Figure out how far left and up.
Move that far left and up from (10,9) to get the 4th vertex.

Or, move that far right and down to get the other choice.

Answer 2

DRAW IT
well, I can put it to the lower right at (x, y)
parallel to top from (5,2):
slope = (9-6)/(10-2) = 3/8
3/8 = (y-2)/(x-5)
3x-15 = 8y-16
8 y = 3 x + 1
==========
parallel to left through (10,9)
slope = (6-2)/(2-5) = -4/3
-4/3 = (y-9)/(x-10)
-4x +40 = 3y -27
3y = -4x + 67
===================
so intersection is vertex
24 y = 9 x + 3
24 y = -32 x + 536
------------------------
0 = 41 x - 533
x = 13
then 3 y = -4 (13) + 67
y = 15/3 = 5
so
(13,5)

Now you do the one on the upper left