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

Question

(1 point)
Responses

rotation of 180° (counterclockwise) about point B
rotation of 180 degrees (counterclockwise) about point upper B

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

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 B maps to point A

Answer 1

To determine the correct rigid motion that would map triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to assess the options provided based on the position and relationships of the triangles' corresponding points.

1. Rotation of 180° (counterclockwise) about point B: This option rotates the triangle around point B. If point B is a corresponding vertex in both triangles, this could be a valid option depending on the configuration.

2. Translation along the vector so that point P maps to point A: This would move point P directly to point A. If point P is a corresponding vertex to point A, this could be valid.

3. Translation along the vector so that point B maps to point Q: Similar to option 2, if point B corresponds to point Q, this translation would be valid.

4. Translation along the vector so that point B maps to point A: This translation would only work if point B corresponds to point A.

To determine the appropriate choice, we need to know which points correspond to each other in both triangles and their current positions. Since we don't have that information, we can assess based solely on typical congruence conditions.

You would typically find that if the triangles are congruent (meaning they are the same size and shape), one of the options should clearly show you how to map fit one triangle onto the other.

If either rotation or translation corresponds to turning one triangle exactly to match the other without distortion, that would be your answer.

Based on the usual conventions in geometric problems using matrices or translations, if the motions correspond to vertex positions correctly (P to A, B to Q, etc.) then those options will hold.

Without more specific information about the angles and the positions of the triangles, it is impossible to conclusively choose the correct answer with just the options provided. However, a common answer might be the one that establishes a direct correspondence between points, such as "translation along the vector so that point P maps to point A" if that relationship is valid based on the triangle's orientation.

If you provide a visual or specific relationships between the points, it would allow for a more definitive answer.