## Parallelograms in the coordinate plane

Construct a figure like this one with Sketchpad using the
following steps.

In the graph menu,

- Choose
Define New Parameter, and make the parameter 1 cm. Don't forget the unit!
- With
the parameter selected on the screen, choose Define Unit Distance.
- Also,
check the item Snap Points.

Then, letting A be the origin, draw segments AB and AC. Construct a parallelogram ABDC.

- Select
all 4 points and choose Measure > Coordinates. If the points have snapped properly,
then the coordinates should all be integers.

- Write
down the coordinates of the points A, B, C, D.

- Also,
measure the equation of line CD.

Now let E be the intersection of
the y-axis and the line CD (i.e., the y-intercept). Construct point F so that ABFE is a
parallelogram.

- Measure
the areas of ABDC and ABFE. How do
they compare. Why?

- Using
the segment AE as the base, what are the measures of the base and height
of ABFE?

### Getting a Formula

Now follow this same construction process but this time
instead of numbers, use letters for the coordinates. Let B = (a,b) and C = (c, d).

- Find
the equation of line CD and the coordinates of E.
- Then
compute the base and height of ABFE and find the area.
- Then
tell how this area compares with the original area of ABDC.

You should have a formula for the area of the parallelogram
ABDC without doing a gruesome calculation involving the height of ABDC because
you found the area of a simpler parallelogram with the same area.
You may find the expression in the formula looks familiar from another
course.