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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

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
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

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
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1 answer

To determine which rigid motion maps triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to analyze the relationship between the points in both triangles.

Given that both triangles are scalene and of equal dimensions, their corresponding sides are equal but their positions in the plane may vary. Rigid motions include translations, rotations, and reflections that preserve size and shape without distorting any angles or side lengths.

Since the question includes pairs of points, we can examine potential translations or rotations that would align the triangles:

  1. Rotation about point B: This option suggests a rotation of 180° about point B. If we visualize this rotation:

    • Point \( P \) would move to a location directly opposite \( B \).
    • Simultaneously, point \( K \) would also get mapped to a location opposite \( B \), which could potentially match point \( L \) if the triangles are appropriately situated.

  2. Translation along the vector so that point B maps to point A: This would shift triangle \( PBK \) so that point \( B \) can coincide with point \( A \). However, the other points would also need to shift correspondingly, and we can't confirm that this results directly in a matching configuration.

  3. Translation along the vector so that point P maps to point A: Similar to the previous translation, without specific orientations, this might not directly lead to triangle congruence depending on other point locations.

  4. Translation along the vector so that point B maps to point Q: This translation would imply bringing \( B \) to a point that is symmetric to where \( Q \) might be located, which could also potentially yield triangle congruence.

Given the provided options, the most likely candidate that maintains the structural integrity of the triangles and their orientations would be the rotation centered at a corresponding point with proper angular fit.

Thus, the best response from the options provided is: rotation of 180° counterclockwise about point B. This allows for a direct mapping from one triangle to another while maintaining their congruency properties.