To determine which rigid motion can map triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to consider the configuration and positions of points in both triangles. However, since the specifics of the positions of these points are not detailed in your question, I will provide a general approach for assessing rigid motions.
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Rotation: If a rotation successfully aligns points \( P \), \( B \), and \( K \) with points \( L \), \( Q \), and the third point of the triangle (which is not specified), it could potentially demonstrate congruence through rotation.
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Translation: A translation would only work if we can shift the entire triangle \( PBK \) such that point \( B \) directly maps to point \( Q \) (or other corresponding points to their positions as necessary).
Let's analyze the response options based on these conditions:
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Rotation of 180° (counterclockwise) about point B: This would map point \( B \) to itself, which could align other points depending on their positions relative to \( B \). You need to check if it maps \( P \) to \( L \) and \( K \) to \( A \).
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Translation along the vector so that point P maps to point A: This may work, assuming both triangles maintain their relative positioning and orientation afterward.
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Translation along the vector so that point B maps to point A: This is unlikely to successfully map \( \triangle PBK \) to \( \triangle LQA \) unless \( A \) is at a location that also aligns \( P \) and \( K \) accordingly—this typically is not sufficient.
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Translation along the vector so that point B maps to point Q: If \( B \) maps to \( Q \), this could allow the other points to map accordingly depending on if angles and lengths are preserved.
Without additional information regarding the relative positions of points \( P, B, K \) and \( L, Q, A \), it is challenging to definitively select one option without knowing how the points are arranged initially.
However, the most promising motions tend to be the first option (rotation of \( 180° \) about point \( B \)) or the last option (translation along the vector so that point \( B \) maps to point \( Q \)).
If you can provide more context or check how the translations align specifically or relate to the rotation, that would help narrow it down further. Usually, one motion will stand out when considering congruence rules, but based on rigid motion, typically you want either a rotation that keeps corresponding sides equal or a translation that does the same.