Problem 22 · 2025 Math Kangaroo
Stretch
Geometry & Measurement
gridspatial-reasoning
On the map shown on the right, we see a city in which there are four schools. Regions A, B, C and D each consist of the points for which the relevant school is closest. The coordinates of the school in region D are \((9\,|\,1)\). What are the coordinates of the school in region A?
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Answer: C — \((1\,|\,5)\)
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Hint 1 of 3
Each border line is the perpendicular bisector of the segment joining two schools, so a school is the mirror image of its neighbour across their shared border.
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Hint 2 of 3
Start from the known D-school at \((9\,|\,1)\) and reflect across the C–D and then A–C borders.
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Hint 3 of 3
Check your candidate: it must be equidistant from the borders of region A and farther from every other school.
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Approach: reflect a known school across the perpendicular-bisector borders
- The three regions meet at the point \((4\,|\,4)\); the A–C border runs along the line through \((4\,|\,4)\) up to \((0\,|\,8)\) and the A–B border down to \((0\,|\,2)\).
- School A must be the reflection of the neighbouring schools across those bisectors, placing it left of and below the corner, at integer coordinates inside region A.
- Testing the options, only \((1\,|\,5)\) is equidistant from both A-borders and closest among all four schools to every point of region A, so A is at \((1\,|\,5)\), choice (C).
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