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.

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.

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

To find the coordinates of the fourth vertex of a parallelogram with three given vertices, we will use the properties of a parallelogram.

A parallelogram has opposite sides that are parallel and equal in length.

Now, from the given three vertices, we can consider two opposite sides of the parallelogram. Let's call the given vertices A(5,2), B(2,6), and C(10,9).

Opposite Side 1: AB
Opposite Side 2: BC

The vector AB can be found by subtracting the coordinates of point A from the coordinates of point B:
AB = B - A = (2-5, 6-2) = (-3, 4).

Similarly, the vector BC can be found by subtracting the coordinates of point B from the coordinates of point C:
BC = C - B = (10-2, 9-6) = (8, 3).

Now, we will add the vector BC to the coordinates of point C to find the coordinates of the fourth vertex:

Coordinates of point C: (10, 9)
Vector BC: (8, 3)

Adding C + BC, we get:
(10, 9) + (8, 3) = (18, 12).

Therefore, the coordinates of the fourth vertex are (18, 12).

To find another point that could be the fourth vertex, we need to find the opposite vector to AB and add it to point A.

The opposite vector of AB can be found by reversing the signs of its coordinates:
Opposite vector of AB: (-(-3), -4) = (3, -4).

Adding this opposite vector to the coordinates of point A:
Coordinates of point A: (5, 2)
Opposite vector of AB: (3, -4)

Adding A + opposite vector of AB, we get:
(5, 2) + (3, -4) = (8, -2).

Therefore, another point that could be the fourth vertex is (8, -2).