The problem involves points $A', B', C'$, which are the images of points $A, B, C$ after a 180-degree rotation around point $O$.

Problem 1: Name a segment whose length is the same as segment $AO$.

Since the points $A', B', C'$ are rotated images, their corresponding distances from point $O$ are preserved after a 180-degree rotation. This means:

• The length of segment $AO$ is equal to the length of segment $A'O$ (because $A'$ is the image of $A$ after a 180-degree rotation).

Thus, segment $A'O$ has the same length as segment $AO$.

Would you like a detailed explanation of the rotation process or have any further questions?

Here are five related questions to deepen your understanding:

1. How do we know the distance from point $O$ to any point remains the same after a 180-degree rotation?
2. What are the other segments in the figure that have equal lengths due to the rotation?
3. How would you verify that the angles between the segments are also preserved under rotation?
4. Why is it important that you do not measure angles or segments directly in the problem?
5. Can you describe how a 180-degree rotation affects the orientation of a triangle?

Tip: A 180-degree rotation around a point maps each point to a position directly opposite across the center of rotation, preserving distances.