# Parallelogram: Coordinate Geometry Setup

- Author:
- Tim Brzezinski

- Topic:
- Coordinates, Geometry, Parallelogram

**BIG POINTS**anywhere you'd like. Interact with the applet below for a few minutes, then answer the questions that follow.

*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? **