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Two acute scalene triangles of equal dimensions and the same orientation. Triangle upper K upper B upper P is in the upper left, and triangle upper A upper Q upper L is in the lower right.

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 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
rotation of 180 degrees (counterclockwise) about point upper B

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

1 answer

To determine which rigid motion would map triangle \( PBK \) to triangle \( LQA \), we need to analyze the relationship between the points in the two triangles:

  1. Identify Corresponding Points:

    • It seems we want to match:
      • \( P \) might correspond to \( L \)
      • \( B \) might correspond to \( Q \)
      • \( K \) might correspond to \( A \)

  2. Evaluate Each Option:

  • Translation along the vector so that point \( B \) maps to point \( A \): This option moves point \( B \) directly to point \( A \), but does not help to arrange the other vertices, \( P \) and \( K \).

  • Translation along the vector so that point \( P \) maps to point \( A \): Moving \( P \) to \( A \) does not match the remaining vertices as \( B \) and \( Q \) would not align correctly.

  • Rotation of 180° (counterclockwise) about point \( B \): This option rotates \( PBK \) around \( B \). If \( B \) functions as the axis of rotation, the other points do not align with \( L \) and \( A \).

  • Translation along the vector so that point \( B \) maps to point \( Q \): This option translates \( B \) directly to \( Q \), which can then lead to verifying that \( P \) will match with \( L \) and \( K \) with \( A\). This translation aligns one pair of triangles properly.

Conclusion: The best option for mapping triangle \( PBK \) to triangle \( LQA \) is:

Translation along the vector so that point \( B \) maps to point \( Q \).