228. The Trapezium Midsegment · half the sum of the parallel sides

Try to break it

Drag A and B to reshape the trapezium. E and F stay at the midpoints of the non-parallel sides, and EF is always parallel to AB and DC, with length exactly (AB + DC)/2 — the average of the two parallel sides. Try to find a shape where EF doesn't match the average; impossible.

How you build it

Build a trapezium and draw its midsegment.

• Point tool: mark point D at the bottom-left.
• Point tool: mark point C to the right of D — DC is the lower parallel side.
• Segment tool: join D to C — the lower parallel side.
• Point tool: mark point A near the top-left, above the base.
• Parallel tool: click A, then click on segment DC. The line through A comes out exactly parallel to DC — that is the top side, no eyeballing needed.
• Point tool: mark point B on that parallel line, to the right of A. Because B sits on the parallel line, AB is automatically parallel to DC.
• Segment tool: join A to D — the left slanting side.
• Segment tool: join B to C — the right slanting side. Trapezium ABCD is complete.
• Midpoint tool: click A then D — E drops at the midpoint of leg AD.
• Midpoint tool: click B then C — F drops at the midpoint of leg BC.
• Segment tool: join E to F — the midsegment. It comes out parallel to AB and DC and equal to ½(AB + DC).

The proof, step by step

Prove that the midsegment of a trapezium equals half the sum of the parallel sides.

• Draw diagonal AC intersecting EF at point G.
• In △ADC, E is midpoint of AD and EG ∥ DC. By Mid-point Theorem, G is midpoint of AC and EG = ½DC.
• In △ABC, G is midpoint of AC and GF ∥ AB. By Mid-point Theorem, GF = ½AB.
• EF = EG + GF = ½DC + ½AB = ½(AB + DC). Hence proved.

Worked example

In a trapezium ABCD, AB ∥ DC. If AB = 12 cm and DC = 8 cm, and E, F are midpoints of AD and BC respectively, find the length of EF.

By the trapezium midpoint theorem, EF = ½(AB + DC) = ½(12 + 8) = 10 cm.

• 8 cm
• 9 cm
• 10 cm — correct
• 11 cm