

Trapezoid (Coordinate Geometry)
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 ycoordinates 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 nonparallel 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:

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.

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.
Things to try

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.

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
Other Coordinate Geometry topics
(C) 2009 Copyright Math Open Reference. All rights reserved

