To determine which rigid motion maps triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to analyze each option given.
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Translation along the vector so that point \( P \) maps to point \( A \) - This implies that point \( P \) is directly moved to point \( A \). In this specific case, after this translation, point \( B \) would also move accordingly, generally leading to a different position from what \( L \) was in triangle \( LQA \).
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Rotation of 180° (counterclockwise) about point \( B \) - A rotation of \( 180° \) about point \( B \) would change the position of points \( P \) and \( K \) effectively. If \( L \) is aligned with the new point that \( P \) rotates to, then it could work, but we can’t confirm without the specific configuration details.
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Translation along the vector so that point \( B \) maps to point \( A \) - Similar to the first option, this translates point \( B \) to point \( A \). However, this may not align the other vertices appropriately unless other relationships hold.
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Translation along the vector so that point \( B \) maps to point \( Q \) - This translates point \( B \) to point \( Q \), which likely realigns the positions of other vertices based on their relative positions.
To select the correct rigid motion, we'd typically rely on additional details about the coordinates or relationships between the points in the triangles. The most appropriate choice usually aligns the corresponding vertices of the triangles accurately.
Based on general understanding without specific coordinates:
- The translation mapping \( B \) to \( Q \) might suggest a reasonable positional change across triangles \( PBK \) and \( LQA \) regardless of specific configurations.
If specific coordinate information were available, further validation could confirm which operation directly aligns the triangles.
Thus, to summarize without explicit triangle vertex locations:
- Among the provided options, Translation along the vector so that point B maps to point Q seems the most plausible choice for mapping \( \triangle PBK \) to \( \triangle LQA \).
For precise answers, verifying with triangle coordinates or properties would yield a more definitive selection.