Use the image to answer the question.

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 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
rotation of 180 degrees (counterclockwise) about point upper B

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

1 answer

To determine which rigid motion maps triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to properly analyze how the points of each triangle correspond to each other.

Given that the triangles are acute, scalene, and in the same orientation while having equal dimensions, a rigid motion must map corresponding vertices of one triangle to the corresponding vertices of the other triangle.

  1. Translation along the vector so that point B maps to point Q: This would move point \( B \) directly to point \( Q \). However, we need to check if the other points also map correctly.

  2. Translation along the vector so that point P maps to point A: This would move point \( P \) to point \( A \). Again, we must check if the other points move correctly.

  3. Rotation of 180° about point B: This would rotate the triangle around point \( B \) but doesn’t directly map \( B \) to \( Q \) or maintain the other triangle's orientation.

  4. Translation along the vector so that point B maps to point A: This would mean moving the triangle down, which does not necessarily align with any mapping from \( PBK \) to \( LQA \).

The most appropriate transformation is translation along the vector so that point B maps to point Q, as that ensures the corresponding vertices align with the definitions of congruent triangles, retaining the dimensions and orientation. Thus, the rigid motion that maps \( \triangle PBK \) to \( \triangle LQA \) is:

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