In trapezium ABCD the side AB is parallel to the side DC, with AB = 12 cm and DC = 26 cm. P is the midpoint of the diagonal AC and Q is the midpoint of the diagonal BD. How long is PQ?

The segment joining the midpoints of the two DIAGONALS of a trapezium is half the DIFFERENCE of the parallel sides: PQ = ½(26 − 12) = 7 cm. Coordinates make it plain — put D(0, 0), C(26, 0), A(a, h) and B(a + 12, h); then P is ((a + 26)/2, h/2) and Q is ((a + 12)/2, h/2), so PQ = (26 − 12)/2 = 7 cm whatever a and h may be. The trap 19 cm is ½(26 + 12), the length of the mid-segment joining the midpoints of the two SLANT SIDES — a different segment, though it lies along the same line. The values 14 cm and 38 cm are the plain difference and the plain sum, each missing the halving.