When a line segment is translated in a plane, the translation involves shifting the segment from one position to another without altering its orientation, length, or shape.
Given these characteristics, let's examine what happens in the context of the specific line segment [tex]\(\overline{AB}\)[/tex] being translated to become [tex]\(\overline{A'B'}\)[/tex].
1. Length Preservation: Translation in the plane keeps the length of segments consistent. It does not stretch, shrink, or alter the dimensions of the object. Therefore, the length of [tex]\(\overline{AB}\)[/tex] remains unchanged after being translated to [tex]\(\overline{A'B'}\)[/tex].
2. True About Lengths: The main property to focus on is that the length of the line segment [tex]\(\overline{A'B'}\)[/tex] will remain equal to the length of the original line segment [tex]\(\overline{AB}\)[/tex]. Hence, [tex]\(AB = A'B'\)[/tex].
Reviewing the provided multiple-choice answers:
- [tex]\(AA = BB\)[/tex]: This equality suggests that the original line segments between A and A, and between B and B, should be equal, which does not make sense in this context.
- [tex]\(A'A' = B'B'\)[/tex]: This statement suggests that the translated line segments between [tex]\(A'\)[/tex] and [tex]\(A'\)[/tex], and between [tex]\(B'\)[/tex] and [tex]\(B'\)[/tex] should be equal, which is also not relevant.
- [tex]\(AA' = BB'\)[/tex]: This implies a relationship between the distances from A to [tex]\(A'\)[/tex] and B to [tex]\(B'\)[/tex], but it is not addressing the length of the translated line segment [tex]\(\overline{A'B'}\)[/tex] in comparison to [tex]\(\overline{AB}\)[/tex].
- [tex]\(AB=BB\)[/tex]: This statement does not hold any logical relevance to the translation scenario at all since it equates the length of [tex]\(\overline{AB}\)[/tex] to itself in a non-meaningful context.
Based on these insights, the correct relationship that captures what remains true after the translation is:
[tex]\[
\boxed{3}
\][/tex]
Thus, the translated line segment [tex]\(\overline{A'B'}\)[/tex] retains the same length as the original line segment [tex]\(\overline{AB}\)[/tex].
