Question
Prove, using vectors, that the line segment joining the mid-points of the non-parallel sides of a trapezium is parallel to the bases and is equal to half the sum of their lenghts.
Solution
Let 
be the position vectors of the vertices A, B, C, D of the trapezium in which AB||CD
Now AB || CD
where 
is some scalar.
Let E be mid-point of AD and F be mid-point of BC.
∴ EF and DC are parallel.
Also EF is parallel to AB as AB is parallel to DC.
Now, 
Hence the result.
