Problem:
Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled , and . What is the sorted order of the four paths from shortest to longest?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
From inspection, we can notice that Path is the shortest path since it avoids going through the entire loop by taking straight-line shortcuts instead of going through the arc portions of the rink. This leaves us with two possible answer choices, and , which only differ in how they sort Path and . To determine which path is longer between Path and , notice that the difference between the two paths can be related by a right triangle as shown here:
The hypotenuse of the triangle was traversed by Sergei in Path while one of the legs is part of Path . By the Pythagorean Theorem, we know that the hypotenuse will always be longer than the legs, and so Path will be longer than Path .
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions