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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.

Sides and diagonals

The lengths of the four sides and two diagonals can be found by using the method described in Distance between two points to find the distance between point pairs.

For example, in the figure above click 'reset' and select "show diagonals' in the options menu. Using the method in Distance between two points, the diagonal AC is the distance between the points A and C:

Similarly the side AB can be found using the coordinates of the points A and B:

Altitude

The altitude of a parallelogram is the perpendicular distance from a vertex to the opposite side (base). In the figure above select "Show Altitude" in the options menu. It will show the altitude from B to the opposite side AB.

The calculate the length of an altitude, we need to find the perpendicular distance from a point to a line. In the above figure we need the distance from B to the line AD.

Simple case

When the chosen base side is exactly horizontal, the altitude is simply the difference is the y-coordinates of B and any point on the base, say A.

In the figure above click 'reset' and select "show altitude' in the options menu. The coordinates of B are (18,26), so its y-coordinate is the second number, 26. Pick a point on the base side, say A. Its y coordinate is 7. The difference between them is the altitude:

The other altitude can be found in the same way if the other sides are exactly vertical, using the x-coordinates of the corresponding points.

Rotated case

If the parallelogram is rotated so that no side exactly vertical or horizontal, then use any of the methods described in Distance from a point to a line.

Area of a parallelogram

Recall from Area of a parallelogram that the area is the altitude times the base. Both of these can be found using the methods shown above.

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 side lengths, diagonals and altitudes of the shape. Click 'show details', 'show diagonals' and 'show altitude' 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 calculations to be slightly off. For example, if you click 'rotated' in the options menu, the shape is the same, yet will show a slightly different altitude. This is because the coordinates round off slightly differently and the altitude is calculated from the displayed coordinates.

For more see Teaching Notes

Other Coordinate Geometry topics