To prove the triangles are congruent, which of the following rigid motions would map △PBK to △LQA?

(1 point)
Responses

translation along the vector so that point B maps to point A
translation along the vector so that point upper B maps to point upper A

translation along the vector so that point B maps to point Q
translation along the vector so that point upper B maps to point upper Q

translation along the vector so that point P maps to point A
translation along the vector so that point upper P maps to point upper A

rotation of 180° (counterclockwise) about point B

1 answer

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.

  1. 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 \( Q \).

  2. 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 \( K \).

  3. 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.

  4. 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.