Question
The triangles are congruent by SSS or HL.
Triangles M N Q and P Q N are connected at side Q N. Triangle M N Q is rotated about point N and then is shifted to the left to form triangle P Q N.
Which transformation(s) can map TriangleMNQ onto TrianglePQN?
translation only
reflection only
rotation, then reflection
rotation, then translation
Triangles M N Q and P Q N are connected at side Q N. Triangle M N Q is rotated about point N and then is shifted to the left to form triangle P Q N.
Which transformation(s) can map TriangleMNQ onto TrianglePQN?
translation only
reflection only
rotation, then reflection
rotation, then translation
Answers
Answered by
GPT-4o mini
To determine which transformations can map triangle \( \triangle MNQ \) onto triangle \( \triangle PQN \), we need to analyze what happens to triangle \( \triangle MNQ \) when it is rotated about point \( N \) and then shifted (translated) to the left to form triangle \( \triangle PQN \).
1. **Rotation about point N**: Rotating triangle \( \triangle MNQ \) about point \( N \) will reposition it in relation to point \( N \). This means the vertices \( M \) and \( Q \) are turned to different locations while \( N \) remains fixed.
2. **Translation**: After the rotation, shifting triangle \( \triangle MNQ \) to the left will align it such that it matches triangle \( \triangle PQN \).
Considering these transformations:
- **Translation only**: This transformation alone cannot suffice, as the initial position of triangle \( \triangle MNQ \) wouldn't align correctly with triangle \( \triangle PQN \) after a simple translation.
- **Reflection only**: Reflection will not fit the described process since it does not involve rotating or translating in the given manner.
- **Rotation, then reflection**: This sequence is not applicable in this case since the problem specifies translating the triangle after rotation, not reflecting it.
- **Rotation, then translation**: This option accurately describes the process of mapping triangle \( \triangle MNQ \) onto triangle \( \triangle PQN \) as it involves the initial rotation about point \( N \) followed by a leftward translation.
Therefore, the correct answer is:
**Rotation, then translation.**
1. **Rotation about point N**: Rotating triangle \( \triangle MNQ \) about point \( N \) will reposition it in relation to point \( N \). This means the vertices \( M \) and \( Q \) are turned to different locations while \( N \) remains fixed.
2. **Translation**: After the rotation, shifting triangle \( \triangle MNQ \) to the left will align it such that it matches triangle \( \triangle PQN \).
Considering these transformations:
- **Translation only**: This transformation alone cannot suffice, as the initial position of triangle \( \triangle MNQ \) wouldn't align correctly with triangle \( \triangle PQN \) after a simple translation.
- **Reflection only**: Reflection will not fit the described process since it does not involve rotating or translating in the given manner.
- **Rotation, then reflection**: This sequence is not applicable in this case since the problem specifies translating the triangle after rotation, not reflecting it.
- **Rotation, then translation**: This option accurately describes the process of mapping triangle \( \triangle MNQ \) onto triangle \( \triangle PQN \) as it involves the initial rotation about point \( N \) followed by a leftward translation.
Therefore, the correct answer is:
**Rotation, then translation.**