Problem 26 · 2018 Math Kangaroo
Stretch
Spatial & Visual Reasoning
reflectionpath-tracing
On an idealised rectangular billiard table with side lengths 3 m and 2 m, a point-shaped ball is pushed away from point M on the long side AB. It is reflected exactly once on each of the other sides, as shown. At what distance from vertex A will the ball hit side AB again if \(BM = 1.2\) m and \(BN = 0.8\) m?
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Answer: E — \(1.8\) m
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Hint 1 of 2
Reflect the path by 'unfolding' the table so the bounces become a straight line.
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Hint 2 of 2
The launch direction is fixed by M and the first bounce point N.
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Approach: unfold the reflections into a straight-line path
- Place A=(0,0), B=(3,0); then M=(1.8,0) (BM=1.2) and the first hit N=(3,0.8) on the short side (BN=0.8), giving direction slope 0.8/1.2 = 2/3.
- Following equal-angle reflections off the right, top and left sides, the unfolded straight path brings the ball back to the long side AB.
- It returns to the point 1.8 m from A.
- Answer: 1.8 m.
Mark:
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