Problem 11 · 2009 Math Kangaroo
Medium
Spatial & Visual Reasoning
path-tracingreflection
A (very small) ball is kicked off from point A on a square billiard table with side length 2 m. After moving along the shown path and touching the sides three times as indicated, the path ends at point B. How long is the path that the ball travels from A to B? (As indicated: angle of incidence = angle of reflection.)
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Answer: B — \(2\sqrt{13}\)
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Hint 1 of 2
Unfold each bounce by reflecting the table, turning the zig-zag into one straight segment.
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Hint 2 of 2
The straight unfolded distance is the hypotenuse of a right triangle whose legs come from the reflections.
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Approach: reflect (unfold) the bounces into a straight line
- Reflecting the square at each bounce straightens the path into a single line from A to the final image of B.
- That line is the hypotenuse of a right triangle with legs 4 and 6 (in units of the 2 m side).
- Its length is √(4² + 6²) = √52 = 2√13.
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