AMC 8 2025 Problem 5
AMC8 2025 年第 5 题
坐标几何★★☆☆☆
Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labled ) and drives to location , then , then , before returning to . What is the shortest distance, in blocks, she can drive to complete the route?
Betty驾驶卡车在一个社区送货，该社区的街道地图如下所示。Betty从工厂（标记为F）出发，依次到达A地、B地、C地，最后返回F地。请问她驾车走完这条路线的最短距离是多少？以街区为单位。
- A.20
- B.23
- C.24
- D.26
- E.28
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
Each shortest possible path from to follows the edges of the rectangle. The following path outlines a path of 24 units:
We can find the shortest distance using a line diagonally from one point to the other, creating a sort of slope, then find the sum of rise and run of the slope, which happens to be the shortest distance, repeat this process until you get back to Point , and you should get 2 + 1 + 3 + 7 + 4 + 2 + 1 + 4, which is equal to 24.
To find the shortest distance Betty drives, we apply Taxicab Geometry (also known as the metric). In a grid system, the distance between any two points and is defined as .
Let the bottom-left corner of the grid be the origin (0,0). Based on the provided map, we assign coordinates to the given locations:
B = (0, 0), C = (2, 4), F = (6, 5), A = (7, 3)
Betty's route follows the sequence . We calculate the Manhattan distance for each leg of the journey:
F (6, 5) to A (7, 3): |6 - 7| + |5 - 3| = 1 + 2 = 3
A( 7, 3) to B (0, 0): |7 - 0| + |3 - 0| = 7 + 3 = 10
B (0, 0) to C (2, 4): |0 - 2| + |0 - 4| = 2 + 4 = 6
C (2, 4) to F (6, 5): |2 - 6| + |4 - 5| = 4 + 1 = 5
The total shortest distance is the sum of these segments: 3 + 10 + 6 + 5 = 24
The shortest distance Betty can drive to complete the route is 24.
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