To find the image of the point $(-5, -7)$ after the transformation $R_{180^\circ} \circ T_{0,0}$, we need to follow these steps:

1. Rotation by $180^\circ$ about the origin:

   • When a point $(x, y)$ is rotated by $180^\circ$ around the origin, the transformation results in the point $(-x, -y)$.

   So, applying the $180^\circ$ rotation to the point $(-5, -7)$: $(-5, -7) \xrightarrow{R_{180^\circ}} (5, 7)$

Thus, the image of the point $(-5, -7)$ after the transformation $R_{180^\circ} \circ T_{0,0}$ is (5, 7).

Would you like more details or clarifications on this transformation?

Here are 5 related questions to explore further:

1. How does the transformation change if we rotate by $90^\circ$ instead of $180^\circ$?
2. What would happen if we applied a translation after the rotation?
3. How does rotating a point by $180^\circ$ affect its distance from the origin?
4. Can you apply the same transformation to any point on the coordinate plane?
5. What is the difference between a rotation around the origin and around another point?

Tip: When rotating points, always pay attention to the center of rotation—it determines how the coordinates change!