AMC 8 2024 Problem 6
AMC8 2024 年第 6 题
面积与周长（综合）★★☆☆☆
Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled P, Q, R, and S. What is the sorted order of the four paths from shortest to longest?
Sergei在滑冰场沿着不同路径滑冰。下图中的灰色线条表示4条不同路径，分别标记为P、Q、R、S。请问这4条路径从短到长的正确排列顺序是什么？
- A.P,Q,R,S
- B.P,R,S,Q
- C.Q,S,P,R
- D.R,P,S,Q
- E.R,S,P,Q
正确答案 · Correct Answer
D
解析 · Solution
解法1
你可以测量路径的长度，直到找到一些可推断出的肯定为真的陈述，例如： 大于 ， 大于 ，且 和 是最小的两个，因此顺序为 $R, P, S, Q.$ 由此我们得到答案, P, S, Q 。显然路径 Q 是最长的路径，其次是 S。因此，剩下路径 P 和 R。注意曲线总是比连接其端点的直线要长，因此路径 P 比路径 R 长。因此，从最短到最长的顺序为 。我们可以比较看起来相似的路径， 与 以及 与 。 比 短，因为它用直线替换了 中的曲线，而直线总是两点之间的最短距离。 中的两条直线可以看作是在冰面上笔直斜线前进的路径，而 中的直线可以看作最初沿冰面斜线前进，然后稍作迂回，最后完成路径。由于两点之间的直线比折线更短，路径 比路径 短。路径 和路径 的唯一区别在于直线部分： 上下移动，而 斜线移动。可以从 的一条线和 的一条线画出一个直角三角形。我们可以看到， 中的直线充当了直角三角形的直角边，而 中的直线则是斜边。直角三角形的直角边总是小于斜边，所以 比 短。综合这些不等关系，答案就是, P, S, Q 。
Solution 1
You can measure the lengths of the paths until you find a couple of guaranteed true inferred statements as such: is greater than , is greater than , and and are the smallest two, therefore the order is $R, P, S, Q.$ Thus we get the answer, P, S, Q. Obviously Path Q is the longest path, followed S. So, it is down to Paths P and R. Notice that curved lines are always longer than the straight ones that meet their endpoints, therefore Path P is longer than Path R. Thus, the order from shortest to longest is . We can compare paths that look similar, with and with . is shorter than because it replaces curved lines in with straight lines, and straight lines are always the shortest distance between two points. The two straight lines in can be seen as paths going diagonally straight across the rink, and the straight lines in can be seen as initially going diagonally across the rink, before taking a short detour and then finishing its path. Since a straight line between two points is shorter than zig lines, path is shorter than path . The only difference between path and path is the straight lines, with going up and down while goes diagonally. A right triangle can be drawn from a line from and a line from . We can see that the straight lines in act as the leg of the right triangle while the hypotenuse is the straight lines from . The legs of a right triangle are always less than the hypotenuse, so is shorter than . Putting together these inequality statements, the answer is, P, S, Q.
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