The student's claim is that a rotation of 180° about the center of a regular polygon carries the polygon onto itself.
To analyze this:
- Regular polygons (such as triangles, squares, pentagons, hexagons, etc.) have symmetrical properties that often include rotational symmetry.
- A rotation of 180° means that every vertex of the polygon is mapped to a vertex directly opposite it (for polygons with an even number of sides) or onto a point in the middle of an edge (for polygons with an odd number of sides).
- For polygons with an even number of sides (like a square or hexagon), the rotation indeed carries the polygon onto itself, as every vertex matches up with another vertex after the rotation.
- For polygons with an odd number of sides (like a triangle or pentagon), the rotation does not carry the polygon onto itself perfectly because there is no vertex that pairs directly opposite another vertex.
Based on this reasoning, the correct statement is:
A. The student's claim is true for all regular polygons.
This statement holds true because, regardless of whether the polygon has an even or odd number of sides, the rotation is symmetrical around the center and results in a mapping that keeps the shape unchanged for the 180° rotation.
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