Congruency of Triangles Quick Check

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Question
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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 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

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
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Answer 1

To determine which rigid motion would map triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to analyze the positions of the points and the nature of the triangles:

1. Triangles are equal and of the same orientation: Since the triangles are congruent and have the same orientation, a rotation or a translation will likely map one triangle to the other.

2. Identifying the points: Let's denote the points:

   • \( P \), \( B \), \( K \) in triangle \( PBK \)
   • \( L \), \( Q \), \( A \) in triangle \( LQA \)

3. Looking for a mapping: We consider the pairs of points:

   • Point \( B \) should map to point \( Q \)
   • Point \( P \) should correspond to point \( A \)
   • Point \( K \) should correspond to point \( L \)

4. Analyzing the responses: We are looking for a rigid motion that will fulfill these mappings:

   • A rotation of 180° about point \( B \) would move point \( P \) to point \( A \) and point \( K \) to point \( L \), while keeping point \( B \) fixed and consequently moving point \( Q \) to the position of \( B \).
   • A translation of any type wouldn't preserve the distances in the way needed for a mapping that swaps positions directly across.

Based on reasoning:

• The best option for mapping \( \triangle PBK \) to \( \triangle LQA \) correctly while preserving the orientation of the triangles is a rotation of 180° counterclockwise about point \( B \), as it aligns the corresponding points correctly.

Thus, the correct response is:

rotation of 180° (counterclockwise) about point B.