Problem 22 · 2007 AMC 8
Hard
Geometry & Measurement
invariantinterior-of-rectangle
A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90° right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
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Answer: C — 5.
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Hint 1 of 2
Resist plotting the exact spot — the 6.2 and the 2 are bait. For any point inside the square, its distance to the left wall plus its distance to the right wall is the full width, 10. Same up-and-down.
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Hint 2 of 2
Hunt for the invariant: when a problem buries you in specific lengths but asks for a sum or average, check whether that quantity stays fixed no matter where the point lands.
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Approach: an inside point's opposite-wall distances always sum to the side
- First confirm the lemming is still inside: it travels 6.2 then 2, well within a 10×10 square. So it's an interior point — the exact spot won't matter.
- Left-wall distance + right-wall distance spans the whole width = 10. Top + bottom likewise = 10. All four distances sum to 20.
- Average = 20 ÷ 4 = 5.
- The payoff: the messy 6.2 and 2 never entered the math — the four distances always total 2×(side), so the average is always side÷2. Spotting that invariant beats any coordinate grind.
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