Problem 27 · 2022 Math Kangaroo
Stretch
Spatial & Visual Reasoning
transformations
A square is placed in a co-ordinate system as shown. Each point \((x\,|\,y)\) of the square is deleted and replaced by the point \(\left(\tfrac{1}{x}\,\middle|\,\tfrac{1}{y}\right)\). Which diagram shows the resulting shape?
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Answer: C
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Hint 1 of 3
Track where the four corners of the square land under the map \((x,y)\to(\tfrac1x,\tfrac1y)\).
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Hint 2 of 3
A straight edge where \(x\) is constant maps to a straight edge (\(1/x\) constant), but an edge where \(x+y\) or a slanted relation holds bends into a hyperbola.
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Hint 3 of 3
Decide whether the transformed sides bow inward or outward to pick the matching picture.
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Approach: image of the square's corners and edges under the reciprocal map
- The corners \((1,1),(2,1),(1,2),(2,2)\) map to \((1,1),(\tfrac12,1),(1,\tfrac12),(\tfrac12,\tfrac12)\), so the image again lives in a small square region near the origin.
- Edges with \(x\) or \(y\) constant stay straight, while the edges along which both coordinates vary become arcs of hyperbolas \(y=c/x\) that curve toward the origin.
- Matching this curved-side small region to the options gives diagram C.
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