Parallelogram (Coordinate Geometry)
 
A quadrilateral with both pairs of opposite sides parallel and congruent, and whose location on the coordinate plane is determined by the coordinates of the four vertices (corners).
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 y-coordinates of any top and bottom point - for example A and B.
  • The width is the difference in x-coordinates 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
    Calculator

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