Question

Betty 开卡车在一个街区送包裹，街道地图如下。Betty 从工厂（标为 $F$ ）出发，依次开到地点 $A$ 、 $B$ 、 $C$ ，然后回到 $F$ 。完成这条路线的最短距离是多少个街区？

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?

$20$

$22$

$24$

$26$

$28$

答案：C

知识点：坐标几何最优化

难度评级：960

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关键是：若在街道网格中从坐标 $(x_1, y_1)$ 开到 $(x_2, y_2)$ ，最短距离为 $|x_2 - x_1| + |y_2 - y_1|$ 。这就是水平街区数加竖直街区数。

从 $F$ 到 $A$ 的最短距离是 $1 + 2 = 3$ 。

从 $A$ 到 $B$ 的最短距离是 $7 + 3 = 10$ 。

从 $B$ 到 $C$ 的最短距离是 $2 + 4 = 6$ 。

从 $C$ 到 $F$ 的最短距离是 $4 + 1 = 5$ 。

相加得到 $3 + 10 + 6 + 5 = 24$ ，选 C。

一个小捷径是注意，从 $B$ 到 $C$ 再到 $F$ 时，经过 $C$ 正好位于从 $B$ 到 $F$ 的某条最短路径上，所以甚至可以先忽略必须停在 $C$ 的要求。

The key idea is that if driving from coordinates $(x_1, y_1)$ to $(x_2, y_2),$ then the shortest distance is $|x_2 - x_1| + |y_2 - y_1|.$ This is often called the Manhattan distance. It is also equal to the number of horizontal blocks between the locations, plus the number of vertical blocks between the locations.

The shortest distance from $F$ to $A$ is then $1 + 2 = 3.$

The shortest distance from $A$ to $B$ is $7 + 3 = 10.$

The shortest distance from $B$ to $C$ is $2 + 4 = 6.$

The shortest distance from $C$ to $F$ is $4 + 1 = 5.$

Adding up all of these numbers, we get $3 + 10 + 6 + 5 = 24,$ which is choice C.

One small possible shortcut for the solution is to notice that when going from $B$ to $C$ to $F,$ the visit to $C$ is conveniently along a shortest path from $B$ to $F$ anyway, so we can even remove the requirement to stop at $C$ from the problem.

2025 AMC 8 第 5 题

答案：C

难度评级：960

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