Would a single point and a fixed distance determine a unique segment in 2-space or 3-space?

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Would a single point and a fixed distance determine a unique segment in 2-space or 3-space like it does in 1-space when given the length of the segment and location of its midpoint? Explain your answer.

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Yes. Any point in any $n$-dimensional space and a fixed $n$-dimensional distance will determine a $n$-dimensional "segment", which in higher dimensions is really just a circle ($n=2$), sphere ($n=3$), or hypersphere ($n>3$).

However, note that in higher dimensions (in euclidean spaces), the word "segment" is reserved to still mean "straight line". You don't call circles segments. In this sense, then no. There are an infinite amount of segments of any given length through a point, due to the reason you have mentioned: the segments can point in any number of directions.