Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labeled $F$) and drives to location $A$, then $B$, then $C$, before returning to $F$. What is the shortest distance, in blocks, she can drive to complete the route?

A. 20

B. 22

C. 24

D. 26

E. 28

1. 2025 AMC 8 官方解析

Answer (C): Traveling from $F$ to $A$ requires Betty to drive 2 blocks south and 1 block east for a total of 3 blocks. Traveling from $A$ to $B$ requires 3 blocks south and 7 blocks west for a total of 10 blocks. Traveling from $B$ to $C$ requires 2 blocks east and 4 blocks north for a total of 6 blocks. Finally, traveling from $C$ back to $F$ requires 4 blocks east and 1 block north for a total of 5 blocks. Therefore the total distance is $3 + 10 + 6 + 5 = 24$ blocks.
To drive from $F$ to $B$, Betty must drive 5 blocks south, and to drive from $A$ to $B$ she must drive 7 blocks west. Because her trip begins and ends at $F$, she must also drive 5 blocks north and 7 blocks east, for a total of $2(5 + 7) = 24$ blocks. Therefore the route cannot be made shorter.

OR

Let $B$ be located at the origin of an $xy$-coordinate grid. Then $A$ is located at $(7, 3)$, $F$ is at $(6, 5)$, and $C$ is at $(2, 4)$. Visiting $F, A, B, C$, and $F$ in order, the total change in the $x$ direction is $1 + 7 + 2 + 4 = 14$ and the total change in the $y$ direction is $2 + 3 + 4 + 1 = 10$. Therefore the total distance Betty drives to complete the route is $14 + 10 = 24$ blocks.