Question
△ABC is congruent to △GHE . Describe a sequence of transformations from △ABC to △GHE that proves their congruence. Try a 90 CW rotation and then a translation.
Answers
GPT-4o mini
To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle GHE \) using a sequence of transformations, you can follow these steps:
1. **90 Degrees Clockwise Rotation**:
Begin by rotating triangle \( \triangle ABC \) 90 degrees clockwise around a point, typically the triangle's centroid or a vertex, depending on how triangle \( \triangle GHE \) is positioned relative to \( \triangle ABC \).
- If \( A \) (or another vertex) is placed at the origin before rotation, a 90-degree clockwise rotation can be achieved using the transformation: \( (x, y) \rightarrow (y, -x) \).
2. **Translation**:
After rotating, you may find that triangle \( \triangle ABC \) has moved to a new position. You can then translate this rotated triangle to align it with triangle \( \triangle GHE \).
- For example, if after rotation, point \( A' \) (the image of A after rotation) needs to move to point \( G \), calculate the translation vector by subtracting the coordinates of \( A' \) from \( G \). This would involve moving each vertex of the rotated triangle by the same vector.
By performing these two transformations, you have successfully moved triangle \( \triangle ABC \) to the position of triangle \( \triangle GHE \), proving that they are congruent through the sequence of a 90 degrees clockwise rotation followed by a translation.
This sequence of transformations demonstrates how to move one triangle to coincide with another, affirming their congruence.
1. **90 Degrees Clockwise Rotation**:
Begin by rotating triangle \( \triangle ABC \) 90 degrees clockwise around a point, typically the triangle's centroid or a vertex, depending on how triangle \( \triangle GHE \) is positioned relative to \( \triangle ABC \).
- If \( A \) (or another vertex) is placed at the origin before rotation, a 90-degree clockwise rotation can be achieved using the transformation: \( (x, y) \rightarrow (y, -x) \).
2. **Translation**:
After rotating, you may find that triangle \( \triangle ABC \) has moved to a new position. You can then translate this rotated triangle to align it with triangle \( \triangle GHE \).
- For example, if after rotation, point \( A' \) (the image of A after rotation) needs to move to point \( G \), calculate the translation vector by subtracting the coordinates of \( A' \) from \( G \). This would involve moving each vertex of the rotated triangle by the same vector.
By performing these two transformations, you have successfully moved triangle \( \triangle ABC \) to the position of triangle \( \triangle GHE \), proving that they are congruent through the sequence of a 90 degrees clockwise rotation followed by a translation.
This sequence of transformations demonstrates how to move one triangle to coincide with another, affirming their congruence.