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

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:

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.

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.

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.

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

- Introduction to coordinate geometry
- The coordinate plane
- The origin of the plane
- Axis definition
- Coordinates of a point
- Distance between two points

- Introduction to Lines

in Coordinate Geometry - Line (Coordinate Geometry)
- Ray (Coordinate Geometry)
- Segment (Coordinate Geometry)
- Midpoint Theorem

- Cirumscribed rectangle (bounding box)
- Area of a triangle (formula method)
- Area of a triangle (box method)
- Centroid of a triangle
- Incenter of a triangle
- Area of a polygon
- Algorithm to find the area of a polygon
- Area of a polygon (calculator)
- Rectangle
- Square
- Trapezoid
- Parallelogram

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