
Parallelogram (Coordinate Geometry)
Try this
Drag any vertex of the parallelogram below. It will remain a parallelogram and its dimensions calculated from its coordinates.
You can also drag the origin point at (0,0).
In coordinate geometry, a parallelogram is similar to an ordinary parallelogram
(See parallelogram definition )
with the addition that its position on the
coordinate plane
is known.
Each of the four vertices (corners) have known
coordinates.
From these coordinates, various properties such as its altitude can be found.
It has all the same properties as a familiar parallelogram:
 Opposite sides are parallel and congruent
 The diagonals bisect each other
 Opposite angles are congruent
See parallelogram definition for more.
Dimensions of a parallelogram
The dimensions of the parallelogram are found by calculating the distance between various corner points.
Recall that we can find the distance between any two points if we know their coordinates.
(See Distance between Two Points )
So in the figure above:
 The height of the parallelogram is the distance between A and B (or C,D).
 The width is the distance between B and C (or A,D).
 The length of a diagonals is the distance between opposite corners, say B and D (or A,C since the diagonals are congruent).
This method will work even if the parallelogram is rotated on the plane, as in the figure above. But
if the sides of the parallelogram are parallel to the x and y axes,
then the calculations can be a little easier.
In the above figure uncheck the "rotated" box to create this condition
and note that:
 The height is the difference in ycoordinates of any top and bottom point  for example A and B.
 The width is the difference in xcoordinates of any left and right point  for example B and D
Example
The example below assumes you know how to calculate the distance between two points, as described in
Distance between Two Points.
In the figure above, click 'reset' and 'show diagonals'
 The height of the parallelogram is the distance between the points A and B. (Using C,D will produce the same result).
Using the formula for the distance between two points, this is
 The width is the distance between the points B and C. (Using A,D will produce the same result).
Using the formula for the distance between two points, this is
 The length of a diagonals is the distance between B and D. (Using A,C will produce the same result).
Using the formula for the distance between two points, this is
Things to try
In the figure at the top of the page, click on "hide details" .
Then drag the corners to create an arbitrary parallelogram.
Calculate the width, height and the length of the diagonals.
Click 'show details' and "show diagonals" to verify your answer.
Limitations
In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place.
This can cause calculatioons to be slightly off.
For more see
Teaching Notes
Other Coordinate Geometry entries
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