Parallelogram: Coordinate Geometry Setup

The applet below contains a parallelogram graphed in the coordinate plane. Notice how 3 of its vertices have variable coordinates. Note that one vertex is fixed at (0,0). You can move the BIG POINTS anywhere you'd like. Interact with the applet below for a few minutes, then answer the questions that follow.
Recall the theorem we've recently proven (using congruent triangles): If a quadrilateral has both pairs of opposite sides congruent, then that quadrilateral is a parallelogram. Use this theorem to help you answer the following questions:


Given that A has coordinates (0,0), B has coordinates (2a, 0), and the green point has coordinates (0, 2c), write expressions (in terms of a, b, and/or c) for the coordinates of points D and C so that quadrilateral ABCD is a parallelogram.


Now, use these variable coordinates to algebraically verify this quadrilateral is a parallelogram by showing, using slopes, that both pairs of opposite sides are parallel. Be sure to label your calculations in the response you type.


In the applet above, select the Midpoint tool, then select point A, then select point C. (This will plot the midpoint of the diagonal with endpoints A and C.)


Given that A = (0,0), use this and one of your results from (1) above to write an expression for the coordinates of the midpoint of AC in terms of a, b, and/or c.


Repeat step (4), but this time use your results from (1) to write an expression (in terms of a, b, and/or c) for the coordinates of the midpoint of segment BD.


Compare your result for (5) with your result from (4). What do you notice? What does your observation tell you about the diagonals of any parallelogram?