Difference between revisions of "2024 AMC 8 Problems/Problem 6"
Blueprimes (talk | contribs) (→Solution 1 (Analysis)) |
|||
| Line 9: | Line 9: | ||
<math>R</math> skips around the boundary of the rink, while <math>P</math> goes around the whole boundary. Hence, the length of path <math>R</math> is less than the length of path <math>P</math>. 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 <math>S</math> is longer than <math>P</math>. Finally, each V-shaped zag path from path <math>Q</math> is longer than a diagonal in path <math>S</math>, so the length of path <math>Q</math> is greater than that of <math>S</math>. Collectively, we obtain the answer <math>\boxed{\textbf{(D)}~R, P, S, Q}</math>. | <math>R</math> skips around the boundary of the rink, while <math>P</math> goes around the whole boundary. Hence, the length of path <math>R</math> is less than the length of path <math>P</math>. 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 <math>S</math> is longer than <math>P</math>. Finally, each V-shaped zag path from path <math>Q</math> is longer than a diagonal in path <math>S</math>, so the length of path <math>Q</math> is greater than that of <math>S</math>. Collectively, we obtain the answer <math>\boxed{\textbf{(D)}~R, P, S, Q}</math>. | ||
| − | ==Video Solution 1(easy to digest) by Power Solve== | + | ==Video Solution 1 (easy to digest) by Power Solve== |
| − | https:// | + | https://www.youtube.com/watch?v=16YYti_pDUg |
Revision as of 17:23, 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 
.
