The problem involves identifying segments that are of the same length as $AB$ in a geometric figure created with a straightedge and compass.

Analysis:

1. The circles are centered at $A$, $B$, and $C$, and were likely constructed with radii equal to other key segments.
2. The segment $AB$ is part of a construction that involves several other equal-length segments, due to the properties of circles and compass constructions.

To identify the segments that are the same length as $AB$:

• $AB$ is a radius of the circle centered at $A$ and the circle centered at $B$.
• Therefore, any other radii of these circles should be of equal length to $AB$.

Segments of equal length to $AB$:

• $BD$: $BD$ is a radius of the circle centered at $B$, so $BD = AB$.
• $BC$: $BC$ is also a radius of the circle centered at $B$, so $BC = AB$.
• $AD$: Since $AD$ is a radius of the circle centered at $A$, $AD = AB$.

Thus, the segments that have the same length as $AB$ are:

• $BD$
• $BC$
• $AD$

The correct selections should be:

• $BD$
• $BC$
• $AD$

Let me know if you'd like more details or further clarification! Here are some related questions you might want to explore:

1. Why are radii of the same circle always equal in length?
2. How can compass and straightedge tools create equal-length segments?
3. What properties of circles make all radii equal?
4. How would you prove that two segments are equal geometrically?
5. Can you identify any other equal-length segments in different geometric constructions?

Tip: Always remember that in geometric constructions using a compass, any segment that is the radius of the same or congruent circles will be equal in length!