In triangle ABC, AB = 8 cm and AC = 14 cm. D is the midpoint of BC and the median AD measures 9 cm. What is the length of BC?

Apollonius' theorem gives $A{B}^2+A{C}^2=2A{D}^2+2B{D}^2$, where $BD$ is HALF of $BC$. So $64+196=2(81)+2B{D}^2$, that is $260=162+2B{D}^2$, giving $B{D}^2=49$ and $BD=7$ cm. $BC$ is twice that, so $BC=14$ cm. The trap 7 cm stops at the half-length $BD$; putting the whole of $BC$ into the theorem instead of half of it also lands on 7 cm; 18 cm simply doubles the median; and 11 cm averages the two given sides.