In trapezium ABCD the side AB is parallel to the side DC, with AB = 9 cm and DC = 15 cm. E is the midpoint of the leg AD and F is the midpoint of the leg BC. The segment EF crosses the diagonal BD at G. What is the length of EG?

EF joins the midpoints of the two legs, so EF ∥ AB ∥ DC. Look at triangle DAB: E is the midpoint of DA and EG ∥ AB, so by the converse of the mid-point theorem G is the midpoint of DB and EG = ½ × AB = 4.5 cm. The remaining piece is GF = ½ × DC = 7.5 cm, and indeed EF = 4.5 + 7.5 = 12 cm, which agrees with the mid-segment value (9 + 15)/2. The trap 7.5 cm halves DC instead of AB — EG is the half lying in the triangle that contains AB.