Problem 6 · 2024 AMC 8
Medium
Geometry & Measurement
spatial-reasoningperimeter
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Answer: D — R, P, S, Q.
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Hint 1 of 2
Don't try to measure anything. Use the plain oval P as your ruler, then ask of each other path: does it cut a corner (shorter) or detour across a diagonal (longer)?
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Hint 2 of 2
Principle: a straight chord across a curve is shorter than the arc; a diagonal across a rectangle is longer than the two sides it skips (hypotenuse > leg). Count cuts vs. diagonals for each path.
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Approach: compare each path to the plain oval boundary, no measuring
- The whole problem is comparisons, not lengths — so compare every path to the plain oval P. R replaces the two rounded ends with straight chords; a straight chord is shorter than the arc it spans, so R < P. R is the shortest, which already narrows you to choices D and E.
- S trades part of the boundary for one diagonal slash across the oval. That diagonal is the hypotenuse of a right triangle, and a hypotenuse is always longer than either leg it replaces — so S > P.
- Q does the same trade but with two crossing diagonals, longer still: Q > S.
- Order shortest→longest: R, P, S, Q — choice D. This transfers: in any "order the lengths" figure problem, look for arcs-vs-chords and diagonals-vs-sides rather than computing — the inequalities decide it.
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