To identify the transformation that maps a regular pentagon with center at (0, 2) onto itself, we must consider the properties of regular polygons. A regular pentagon has five equal sides and five equal angles, and its symmetry allows for several transformations that can leave it invariant.
One of the key transformations that can map the pentagon onto itself is a rotation about its center. For a regular pentagon, these rotations can be by multiples of 72 degrees (i.e., 0°, 72°, 144°, 216°, and 288°). Since the center of our pentagon is at (0, 2), we can visualize or perform these rotations around that point.
In addition to rotation, an appropriate reflection could also map the pentagon onto itself. Reflections can occur over axes that either pass through vertices or midpoints of the sides of the pentagon.
Therefore, the transformations that map the regular pentagon onto itself with center (0, 2) are:
- Rotations by 72°, 144°, 216°, and 288°.
- Reflections over appropriate lines of symmetry.
This effectively demonstrates that both rotations and reflections related to the pentagon’s symmetries sustain its integrity and orientation, allowing it to map onto itself perfectly.