Question

Sergei 沿不同路线绕冰场滑行。下图中标出的线显示了四条路线，分别标为 P、Q、R、S。按从短到长排列，这四条路线的顺序是什么？

Sergei skated around an ice rink, gliding along different paths. The marked 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?

P, Q, R, S

P, R, S, Q

Q, S, P, R

R, P, S, Q

R, S, P, Q

答案：D

知识点：弧极限情形界定

难度评级：1070

小提示：

比较路径片段：弧线比对应弦更长

Compare the path pieces: arcs are longer than their chords

大提示：

选项已经一致认为路线 R 最短、路线 Q 最长

The answer choices already agree that Path R is shortest and Path Q is longest

视频讲解：

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文字解答：

路线 R 最短，因为它用直线弦替代了弧线部分。路线 Q 最长，因为它在仍然使用弯曲端部的同时，还包含最多的内部穿越距离。

接下来只需比较路线 P 和 S。路线 S 中相关的直线段是横穿冰场的一条对角线，而路线 P 中对应的直线段是同一个直角三角形的一条直角边。对角线比直角边长，所以路线 S 比路线 P 长。

从短到长的顺序为 R、P、S、Q。

所以正确答案是 D。

Path R is shortest because it replaces curved arc portions with straight-line chords. Path Q is longest because it includes the most interior crossing distance while still using the curved ends.

It remains to compare paths P and S. The relevant straight piece in path S is a diagonal across the rink, while the corresponding straight piece in path P is a side of the same right triangle. A diagonal is longer than a side, so path S is longer than path P.

The order from shortest to longest is R, P, S, Q.

Thus, D is the correct answer.

2024 AMC 8 第 6 题