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 four 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

You can measure the lengths of the paths until you find a couple of guaranteed true inferred statements as such: Q is greater than S, P is greater than R, and R and P are the smallest two, therefore the order is R, P, S, Q. Thus we get the answer $\boxed{\textbf{(D)}~R, P, S, Q}$ .

- U-King

~ cxsmi (minor $\LaTeX$ edits)

Solution 2 (Intuitive)

Obviously Path Q is the longest path, followed by Path S.

So, it is down to Paths P and R.

Notice that curved lines are always longer than the straight ones that meet their endpoints, therefore Path P is longer than Path R.

Thus, the order from shortest to longest is $\boxed{\textbf{(D) } \text{R, P, S, Q}}$ .

~MrThinker

Video Solution 1 by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc

~NiuniuMaths

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=ZedvqIYTDG3D20Rp&t=1301

~Math-X

Video Solution by Power Solve (easy to digest!)

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

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=386

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search