## Midpoint trapezium (trapezoid) theorem generalized

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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.

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A trapezium theorem generalized

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

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Midpoint Hexagon Result

In the sketch above, *G* and *H* are the respective midpoints of the opposite sides *AB* and *DE* of the hexagon *ABCDEF*. Measurements for *GH* and (*AF* + *FE* + *BC* + *CD*)/2 are given.

Drag any of the vertices of *ABCDEF* and carefully compare the measurements of *GH* and (*AF* + *FE* + *BC* + *CD*)/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.

Can you use the first result for a quadrilateral to *explain why* (prove that) your hexagon conjecture is true? Can you generalize further to an octagon or decagon? Check your explanation (proof) by clicking here.

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Modified by Michael de Villiers, 14 June 2014.