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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.
A trapezium theorem generalized
Check your conjecture above by clicking here. Can you explain why (prove that) your conjecture is true?
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.
Modified by Michael de Villiers, 14 June 2014.