2024 AMC 8 Problem 6
Attempt Problem 6 of the 2024 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
6.
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
Answer: D
Small hint:
Compare the path pieces: arcs are longer than their chords
Big hint:
The answer choices already agree that Path R is shortest and Path Q is longest
Video solution:
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Written solution:
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.
Problem 6 in other years
- 1985 AMC 8
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