Shortest Distances (2D)

Geometry Level pending

What is the length of the shortest distance between the center of the left side and the bottom-right corner of this 44 x 88 pool table, given that you have to touch the top and bottom side at least once, and the bottom-right corner doesn't count? (An example path is shown above)

50 5 50\sqrt{5} 88 2 88\sqrt{2} 135 44 3 44\sqrt{3} 22 41 22\sqrt{41}

Charles Z by Charles Z , 14, USA

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1 solution

We can treat the path of the pool ball as a beam of light and the top and bottom edges of the pool table as mirrors. Then the reflection of the path continues as a straight line as shown in the figure. The shortest path of the pool ball is a straight line from the midpoint of the left edge to the right-bottom corner and its length is ℓ = 11 0 2 + 8 8 2 = 22 41 \ell = \sqrt{110^2+88^2} = \boxed {22\sqrt{41}} .

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