Problem 19 · 2024 Math Kangaroo
Hard
Spatial & Visual Reasoning
net-folding
Otis builds the net of a solid from squares and triangles, as shown; every side of the squares and triangles has length 1. He folds the net to form the solid shown. What is the distance from A to B?
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Answer: A — \(1+\sqrt{2}\)
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Hint 1 of 2
Folding the band of squares makes a square tube; the triangles cap it into the solid, and every edge has length 1.
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Hint 2 of 2
Give the corners simple 3-D coordinates, then A and B are two of those corners; use the distance formula.
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Approach: fold to coordinates, then apply the distance formula
- The four squares fold into the sides of a square prism of side 1, and the triangles fold over to close it, so all vertices sit on a unit grid.
- Placing the vertices on coordinates, A and B land so that they differ by 1 in one direction and by a face diagonal \(\sqrt2\) lined up in the same straight line.
- Adding those aligned pieces gives the straight-line distance \(AB = 1+\sqrt2\), answer A.
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