# 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:

## 1.

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.

## 2.

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.

## 3.

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.)

## 4.

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.

## 5.

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.

## 6.

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?