Question
In 3–5 sentences, explain how a regular pentagon rotations onto itself onto itself. State all possibilities and show work to justify your answer.
Answers
GPT-4o mini
A regular pentagon can rotate onto itself in five distinct ways. The rotations occur at angles of \(0^\circ\), \(72^\circ\), \(144^\circ\), \(216^\circ\), and \(288^\circ\).
1. **\(0^\circ\)** rotation is the identity, where the pentagon remains unchanged.
2. **\(72^\circ\)** rotation moves each vertex to the position of the next vertex clockwise.
3. **\(144^\circ\)** rotation shifts each vertex two positions clockwise.
4. **\(216^\circ\)** rotation moves each vertex three positions clockwise.
5. **\(288^\circ\)** rotation shifts each vertex four positions clockwise.
These rotations can be verified by noting that each vertex will land on a vertex of the pentagon, thus maintaining the shape and symmetry. This symmetry implies that the regular pentagon has five rotational symmetries.
1. **\(0^\circ\)** rotation is the identity, where the pentagon remains unchanged.
2. **\(72^\circ\)** rotation moves each vertex to the position of the next vertex clockwise.
3. **\(144^\circ\)** rotation shifts each vertex two positions clockwise.
4. **\(216^\circ\)** rotation moves each vertex three positions clockwise.
5. **\(288^\circ\)** rotation shifts each vertex four positions clockwise.
These rotations can be verified by noting that each vertex will land on a vertex of the pentagon, thus maintaining the shape and symmetry. This symmetry implies that the regular pentagon has five rotational symmetries.