To understand the transformation that maps a regular pentagon onto itself, particularly one centered at the point (0, 2), we need to explore the geometric properties of regular polygons.
A regular pentagon has rotational and reflectional symmetries. For a pentagon, the notable transformations include:
- Rotations: The pentagon can be rotated about its center. In our case, since the center is at (0, 2), the angles of rotation that map the pentagon onto itself are 72°, 144°, 216°, 288°, and 360° (which returns to the starting position).
- Reflections: There are five lines of symmetry in a regular pentagon. Each line passes through a vertex and the midpoint of the opposite side. Reflecting across these lines will also map the pentagon onto itself.
In combination, both rotations and reflections can create various transformations that keep the pentagon’s structure intact. So, the transformations that will map a regular pentagon centered at (0, 2) onto itself include the set of rotations by multiples of 72° and reflections across the five lines of symmetry.
To conclude, using these symmetrical properties, you can achieve multiple mappings of the pentagon onto itself, thereby preserving its geometric integrity and visual appeal.