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 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
translation along the vector so that point B
maps to point A
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 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
translation along the vector so that point B
maps to point A
Answers
GPT-4o mini
To determine which rigid motion would map triangle \( PBK \) to triangle \( LQA \), let's consider the relationships between the points in both triangles.
1. **Translation**: This method would involve moving points directly from their current locations to new locations without rotating them. For example, translating \( P \) to \( A \) or \( B \) to \( Q \) would not necessarily align the triangles correctly unless the orientation and shapes were identical, which in this case they are not likely to be in a scalene triangle.
2. **Rotation**: To understand if a rotation fits, we can evaluate rotating about a center point (in this case, point \( B \)). A 180-degree rotation implies that each point moves to a position that is directly opposite it through point \( B \).
However, the comment provides multiple options for translations that seem to suggest that in one of them, if \( B \) is translated to \( Q \), it would suggest that both triangles maintain their proportional lengths and angles despite their original positions.
Given that both triangles are described as being acute and scalene, they have uniquely defined angles and sides, indicating that although they share a similar shape, they must align through specific translations or rotations about the point \( B \).
Considering these factors, the accurate rigid motion that would map triangle \( PBK \) to triangle \( LQA \) appears to be:
**Translation along the vector so that point B maps to point Q.**
This would allow the two triangles to align correctly while maintaining their orientation and congruency since they are described as being of equal dimensions and shapes.
So the conclusion is:
**Translation along the vector so that point B maps to point Q.**
1. **Translation**: This method would involve moving points directly from their current locations to new locations without rotating them. For example, translating \( P \) to \( A \) or \( B \) to \( Q \) would not necessarily align the triangles correctly unless the orientation and shapes were identical, which in this case they are not likely to be in a scalene triangle.
2. **Rotation**: To understand if a rotation fits, we can evaluate rotating about a center point (in this case, point \( B \)). A 180-degree rotation implies that each point moves to a position that is directly opposite it through point \( B \).
However, the comment provides multiple options for translations that seem to suggest that in one of them, if \( B \) is translated to \( Q \), it would suggest that both triangles maintain their proportional lengths and angles despite their original positions.
Given that both triangles are described as being acute and scalene, they have uniquely defined angles and sides, indicating that although they share a similar shape, they must align through specific translations or rotations about the point \( B \).
Considering these factors, the accurate rigid motion that would map triangle \( PBK \) to triangle \( LQA \) appears to be:
**Translation along the vector so that point B maps to point Q.**
This would allow the two triangles to align correctly while maintaining their orientation and congruency since they are described as being of equal dimensions and shapes.
So the conclusion is:
**Translation along the vector so that point B maps to point Q.**