Problem 8 · 2011 Math Kangaroo
Medium
Geometry & Measurement
transformationsspatial-reasoning
The two bold lines shown are rotations of each other. Which of the given points could be the centre of this rotation?
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Answer: C — X and T
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Hint 1 of 3
A rotation centre must be the same distance from each endpoint of a segment as from the matching endpoint of its image.
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Hint 2 of 3
Equivalently, the centre lies on the perpendicular bisector of the segment joining each point to where it lands.
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Hint 3 of 3
Test each labelled grid point to see which send the vertical bold segment exactly onto the horizontal one.
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Approach: the centre lies on the perpendicular bisectors joining matched endpoints
- Under a rotation, each point of the first segment moves to a point of the second, and the centre is equally far from both.
- So the centre must lie on the perpendicular bisector of the line joining each endpoint to its image; the true centre lies on all of them at once.
- Checking the four marked points on the grid, both X and T sit on those bisectors and turn the vertical bold segment onto the horizontal one (by ±90°).
- So both X and T work, choice (C).
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