To determine which rigid motion would map triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we should consider the corresponding points of each triangle:
- Points in \( \triangle PBK \) are \( P, B, \) and \( K \).
- Points in \( \triangle LQA \) are \( L, Q, \) and \( A \).
Rigid motions include translations, rotations, and reflections. From the options given, let's analyze them one by one based on the mapping of the vertices.
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Translation along the vector so that point B maps to point A: This would move \( B \) to \( A \). However, we would need to know where \( P \) and \( K \) would map, which does not guarantee they correspond to \( L \) and \( Q \).
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Translation along the vector so that point B maps to point Q: This would move \( B \) to \( Q \). As with the previous option, we cannot confirm the corresponding mapping of \( P \) and \( K \).
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Translation along the vector so that point P maps to point A: Again, moving \( P \) to \( A \) doesn't provide information about how \( B \) and \( K \) correspond.
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Rotation of 180° (counterclockwise) about point B: This would rotate the entire triangle about point \( B \).
- If \( B \) remains in place, \( P \) would move to some point \( P' \) on the opposite side of \( B \), likely corresponding to \( L \), and \( K \) would move to some point \( K' \) which would correspond to \( Q \). This rotation maintains distances and angles, consistent with triangle congruence.
Therefore, the best option to map \( \triangle PBK \) to \( \triangle LQA \) is:
Rotation of 180° (counterclockwise) about point B.