Parallelograms in the coordinate plane
Construct a figure like this one with Sketchpad using the
In the graph menu,
Define New Parameter, and make the parameter 1 cm. Don't forget the unit!
the parameter selected on the screen, choose Define Unit Distance.
check the item Snap Points.
Then, letting A be the origin, draw segments AB and AC. Construct a parallelogram ABDC.
all 4 points and choose Measure > Coordinates. If the points have snapped properly,
then the coordinates should all be integers.
down the coordinates of the points A, B, C, D.
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
the areas of ABDC and ABFE. How do
they compare. Why?
the segment AE as the base, what are the measures of the base and height
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).
the equation of line CD and the coordinates of E.
compute the base and height of ABFE and find the area.
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