2024 AMC 8 Problems/Problem 6

Problem

Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show foru of the paths labeled $P$ , $Q$ , $R$ , and $S$ . What is the sorted order of the four paths from shortest to longest?

$\textbf{(A)}\ P,Q,R,S \qquad \textbf{(B)}\ P,R,S,Q \qquad \textbf{(C)}\ Q,S,P,R \qquad \textbf{(D)}\ R,P,S,Q \qquad \textbf{(E)}\ R,S,P,Q$

Solution 1 (Analysis)

$R$ skips around the boundary of the rink, while $P$ goes around the whole boundary. Hence, the length of path $R$ is less than the length of path $P$ . Now, using the fact that the hypotenuse of a right triangle is greater than both of its legs, it is clear that the path described in $S$ is longer than $P$ . Finally, each V-shaped zag path from path $Q$ is longer than a diagonal in path $S$ , so the length of path $Q$ is greater than that of $S$ . Collectively, we obtain the answer $\boxed{\textbf{(D)}~R, P, S, Q}$ .

Video Solution 1 (easy to digest) by Power Solve

https://www.youtube.com/watch?v=16YYti_pDUg

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2024 AMC 8 Problems/Problem 6

Problem

Solution 1 (Analysis)

Video Solution 1 (easy to digest) by Power Solve