Trapezoid (Coordinate Geometry)

A quadrilateral that has one pair of parallel sides,
and where the vertices have known coordinates.
Try this Drag any vertex of the trapezoid below. It will remain a trapezoid. You can also drag the origin point at (0,0).

As in plane geometry, a trapezoid is a quadrilateral with one pair of parallel sides. (See Trapezoid definition). In coordinate geometry, each of the four vertices (corners) also have known coordinates.

Altitude of a trapezoid

In the figure above, click on 'reset' then 'show altitude'. The altitude is the perpendicular distance between the two bases (parallel sides). To find this distance, we can use the methods described in Distance from a point to a line. For the point, we use any vertex, and for the line we use the opposite base. In the figure above we have used the distance from point B to the opposite base AD.

This method will work even if the trapezoid is rotated on the plane, but if the sides of the trapezoid are parallel to the x and y axes, then the calculations can be a little easier. The altitude is then the difference in y-coordinates of any point on each base, for example A and B.

Median of a Trapezoid

In the figure above, click on 'show median'. Recall from Median of a Trapezoid that the median is a line segment linking the midpoints of the two legs of the trapezoid. (The legs are the two non-parallel sides.) We can find the midpoint of a leg by using the method described in Midpoint of a line segment. By applying this twice, once for each leg, the median can be drawn between them.

The length of the median can be found in two ways:

1. The median length is the average of the two bases (parallel sides). Find the length of each base by using the method described in Distance between Two Points. Then find the average of these two lengths by adding them and dividing by 2.
2. Find the midpoints of the legs using the method described in Midpoint of a line segment, then find the distance between them as described in Distance between Two Points.

Example

In the worked examples below, we will calculate the properties of the trapezoid in the figure above. Press 'reset' first.
• The altitude of the trapezoid.
Since in this case the bases (parallel sides) of the trapezoid are parallel to the x-axis, the altitude can be found as the difference between the y-coordinates of any point on each base. Let's pick B and A. The y-coordinate of B is 31, and the y coordinate of A is 7, so:
Altitude = 31–7 = 24
Click on 'show altitude' to verify. If the trapezoid had been rotated, then the altitude would be found using Distance from a point to a line, in this case using say the point A, and finding the distance to the line BC.
• The endpoints of the median are located at the midpoints of AB and CD, which can be found using the method described in Midpoint of a line segment. To find G, the midpoint of AB:
The x-coordinate of G is the average of the x-coordinates of A and B: and the y coordinate is the average of the y-coordinates of A and B: and so, one end of the median is at G(11,19). By the same method, the other end is at H(42,19). Click 'show median' to verify.
• The length of the median is the distance between the midpoints G,H of AB and CD. Using the method described in Distance between Two Points, we see that in this case, the median is parallel to the x-axis, so the length is the difference in the x-coordinates of G and H:
Median Length = 42–11 = 31

Things to try

1. In the figure at the top of the page, click on "hide details" . Then drag the corners to create an arbitrary trapezoid. Calculate the altitude, and the location and length of the median. Click 'show details' to verify your answer.
2. Repeat with a rotated trapezoid by clicking on 'rotated'.

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

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref