Question

the parallel sides of a trapezium are p and q respectively prove that the line joining the midpoints of its non parallel sides is half p + q

Answer 1

Step-by-step explanation:

Let's say the (position vectors of) the vertices of the trapezium are a, b, c, d, in order around the perimeter, with p corresponding to the side from a to b, and q corresponding to the side from D to C.

Then p = b - a and q = c - d.

The midpoints of the non-parallel sides are

( a + d ) / 2  and  ( b + c ) / 2.

The vector from one midpoint to the other is then

( b + c ) / 2  -  ( a + d ) / 2

= ( b + c - a - d ) / 2

= ( ( b - a ) + ( c - d ) ) / 2

= ( p + q ) / 2