In triangle ABC the median AD is drawn to the side BC, and G is the centroid. The line through G parallel to AB meets BC at the point X. If BC = 18 cm, what is the length of BX?

D is the midpoint of BC, so BD = DC = 9 cm, and the centroid divides the median with AG : GD = 2 : 1, which means DG : DA = 1 : 3. Now look at triangle DAB: the line through G parallel to AB cuts DA at G and DB at X, so by the basic proportionality result DX : DB = DG : DA = 1 : 3. Hence DX = 3 cm and BX = BD − DX = 9 − 3 = 6 cm. The trap 3 cm answers DX, the piece next to D; 9 cm answers BD, forgetting the parallel line altogether; and 12 cm answers XC = DX + DC, measuring from the wrong end.