To determine which figure is the image of $R$ after a $180^\circ$ rotation about its center, we need to consider how a figure behaves under a $180^\circ$ rotation.

When a figure is rotated by $180^\circ$ around a point, every point on the figure moves to a new position directly opposite to its original position relative to the center of rotation. This means that after the rotation, the image of each point will be the reflection of the original point across the center, flipped halfway around the center.

Without seeing the actual image of $R$, I can give you a general way to approach this:

1. Identify the center of rotation.
2. Measure the distance and direction from the center of rotation to a point on figure $R$.
3. Rotate each point on figure $R$ by $180^\circ$, meaning you would move each point to the opposite side of the center, at the same distance.

Since you have choices (A, B, C, D), look for the figure that matches this description.

Would you like to upload an image of the figure $R$ so that I can provide a more specific answer? Or do you have any other details to share?