To determine which rigid motion maps triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to analyze the options provided.
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Translation along the vector so that point B maps to point Q:
- This would translate point B to point Q, but we need to consider whether this mapping aligns the entire triangles correctly. Depending on the locations of points P, K, L, and A, this translation might not map the triangles onto each other.
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Translation along the vector so that point B maps to point A:
- This would move point B to point A. If point P and point Q are positioned such that, after this translation, they coincide, and the same goes for point K and point L, then this could be a valid mapping. However, without more information about the relative positions, we can't confirm this.
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Rotation of 180° (counterclockwise) about point B:
- This would effectively flip the triangle around point B. Whether this correctly maps \( \triangle PBK \) to \( \triangle LQA \) depends on the positions of points P, B, K relative to L, Q, A, respectively. There’s no confirmation that this is the correct motion without a visual.
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Translation along the vector so that point P maps to point A:
- Similar to the previous translations, moving point P to point A would alter the configuration of the triangle and might not correctly align it with \( \triangle LQA \) without additional information.
In conclusion, without specific information about the configuration of points P, B, K and L, Q, A, it's challenging to definitively determine which rigid motion is correct. However, if we are looking for a rigid motion that directly translates or rotates one triangle to align with the other, the second option (translation so that point B maps to point A) often has the strongest potential to map the relative configurations of points provided that the remaining points align accordingly.
Recommendation: Option (2) "translation along the vector so that point B maps to point A" seems the most plausible choice without additional context.