When a line segment [tex]\(\overline{AB}\)[/tex] is translated in a plane to form [tex]\(\overline{A'B'}\)[/tex], the geometric properties of the segment remain unchanged. Translation is a rigid motion, meaning it preserves the original length and orientation of the line segment. Let's analyze the options provided:
1. Option 1: [tex]\(AA = BB\)[/tex]
This option does not make sense in the context of translating a line segment, as it seems to indicate that each endpoint [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is identical before and after the translation, which is incorrect because [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are moved to new positions.
2. Option 2: [tex]\(A'A' = B'B'\)[/tex]
This is also incorrect because it suggests that the points [tex]\(A'\)[/tex] and [tex]\(A'\)[/tex], as well as [tex]\(B'\)[/tex] and [tex]\(B'\)[/tex], should have the same coordinates or something similar, which is not relevant to the translation itself.
3. Option 3: [tex]\(AA' = BB'\)[/tex]
This suggests the distances between corresponding original and translated points are equal. While the distance moved during a translation could be equal, it's not the defining characteristic of the relationship between [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{A'B'}\)[/tex].
4. Option 4: [tex]\(AB = A'B'\)[/tex]
This option correctly states that the length of line segment [tex]\(\overline{AB}\)[/tex] is equal to the length of the translated line segment [tex]\(\overline{A'B'}\)[/tex]. Translation does not change the length or the orientation of the segment, only its position.
Therefore, the correct answer is:
[tex]\[ \overline{AB} = \overline{A'B'} \][/tex]
This succinctly captures the invariant property of the line segment under translation in a plane: its length remains constant.
