## Midpoint trapezium (trapezoid) theorem generalized

NOTE: Please WAIT while the applets below load. If a security pop-up menu appears in your browser, please choose RUN/ALLOW to let the applets run properly. It is completely safe & can be trusted. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applets loading, please go here for additional information on Java settings that should resolve the issue.

A well known theorem for a trapezium that appears in many school textbooks is the following: Given any trapezium ABCD with AD // BC and E and F the respective midpoints of opposite sides AB and CD. Then EF = (AD + BC)/2.

Below is given a general quadrilateral ABCD with E and F the respective midpoints of opposite sides AB and CD, and measurements given for EF and (AD + BC)/2.

Drag any of the vertices of ABCD and carefully compare the measurements of EF and (AD + BC)/2. What do you notice? What conjecture can you make?

Ensure that you check your conjecture carefully, and also look at concave and crossed cases as well.

Check your conjecture above by clicking here. Can you explain why (prove that) your conjecture is true?