CEMC Hypatia, 2010

Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $3000$ km from $A$ and with $B$ a distance of $3250$ km from $A$. To take a bus, it costs Piravena $0.15 per kilometre. To take an airplane, it costs her a $100 booking fee, plus $0.10 per kilometre. Piravena chose the least expensive way to travel between cities and her total cost was $1012.50. Given that she flew from $A$ to $B$, determine her method of transportation from $B$ to $C$ and her method of transportation from $C$ to $A$.