In triangle ABC the bisector of ∠A meets BC at D, and measurement gives BD = 4.5 cm and DC = 6 cm. If AB = 6 cm, what is the length of AC?

The angle-bisector theorem says AB : AC = BD : DC. So 6 : AC = 4.5 : 6, that is 4.5 × AC = 36 and AC = 8 cm. A sanity check: DC is the longer of the two pieces, so AC must be the longer of the two sides, and 8 > 6. Writing the proportion upside down (6 : AC = 6 : 4.5) gives the trap 4.5 cm, and 10.5 cm is simply BC.