Difference between revisions of "2024 AMC 8 Problems/Problem 6"
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| + | ==Problem== | ||
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| + | Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show foru of the paths labeled <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math>. What is the sorted order of the four paths from shortest to longest? | ||
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| + | <math>\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</math> | ||
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==Solution 1 (Analysis)== | ==Solution 1 (Analysis)== | ||
Revision as of 18:14, 25 January 2024
Problem
Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show foru of the paths labeled 
, 
, 
, and 
. What is the sorted order of the four paths from shortest to longest?
Solution 1 (Analysis)
skips around the boundary of the rink, while goes around the whole boundary. Hence, the length of path is less than the length of path . 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 is longer than . Finally, each V-shaped zag path from path is longer than a diagonal in path , so the length of path is greater than that of . Collectively, we obtain the answer 
.
