The problem asks to name a segment whose length is the same as segment $AO$.

Explanation:

Given the points $A', B', C'$ are the images of the 100-degree rotations of points $A, B, C$ respectively around point $O$, and we see that $\angle AOB = 79^\circ$ and $\angle BOC = 35^\circ$:

1. Rotations preserve the distance from the center of rotation. This means that for any point and its rotated image, the distance to the center remains the same.

2. Therefore, the segment $A'O$ should have the same length as $AO$, $B'O$ should have the same length as $BO$, and $C'O$ should have the same length as $CO$.

Conclusion:

The segment whose length is the same as $AO$ is $A'O$.

Would you like more details or have any questions?

Here are five related questions you might find interesting:

1. What is the significance of rotational symmetry in geometry?
2. How does the concept of rotational symmetry apply to other geometric transformations?
3. Can the rotation angle affect the length of segments in any geometric transformation?
4. What happens to the orientation of figures after a rotation by different angles (e.g., 90 degrees, 180 degrees)?
5. How would the problem change if the rotation was about a different point instead of point $O$?

Tip: When dealing with rotations, always remember that they preserve the distances between the center of rotation and any point on the figure.