To determine which rigid motion would map triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to consider how the points in triangle \( PBK \) relate to the points in triangle \( LQA \).
-
Rotations: Checking the first two options about rotating 180° about point B:
- A rotation of 180° around point B would map point P to a point directly opposite it across B. This does not align points A and P correctly, so it would not map \( \triangle PBK \) to \( \triangle LQA \).
-
Translations: Considering the translations:
- Translating along the vector that maps point P to point A means we would move point P to where point A is located. This sets \( P \to A \), but we need to check that the other points will also align correctly afterward.
- Translating point B to point Q: This would send point B directly to where Q is located. This could work depending on the placement of the triangles.
After thorough consideration, the most critical step is to determine if we can find a translation that directly maps the entire triangle accurately while ensuring that the relative lengths and orientations stay consistent:
Looking at the last option:
- Translation along the vector so that point B maps to point A: This translation directly moves point B in triangle \( PBK \) to point A in triangle \( LQA \).
If \( B \) maps to \( A \), we must consider the positioning of the entire triangle. Since \( \triangle PBK \) and \( \triangle LQA \) are scalene and have identical shapes, we can assume the translation aligns correctly in this case.
Thus, the correct answer is: Translation along the vector so that point B maps to point A.