To show that triangles \( \triangle ABC \) and \( \triangle GHE \) are congruent through a sequence of transformations, we can apply a combination of translations, rotations, and reflections. Here’s a step-by-step description of a possible sequence of transformations:
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Translation: Start by translating \( \triangle ABC \) so that point \( A \) coincides with point \( G \). This involves moving the entire triangle a certain distance in the plane (up, down, left, or right) until point \( A \) is at the same coordinates as point \( G \).
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Rotation: Next, rotate the triangle \( \triangle ABC \) around point \( G \) so that point \( B \) aligns with point \( H \). The angle of rotation will need to be determined based on the relative positions of points \( B \) and \( H \).
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Reflection (if necessary): If, after the rotation, point \( C \) does not align with point \( E \), you may need to reflect the triangle over a line that goes through point \( G \) (or the line that corresponds to the orientation of the triangles).
Each of these transformations maintains the distances and angles of the triangle, leading to congruence. The final result will position \( \triangle ABC \) exactly over \( \triangle GHE \), thus proving their congruence.
Summary
In summary, the transformations are:
- Translate \( \triangle ABC \) so \( A \) aligns with \( G \).
- Rotate \( \triangle ABC \) around point \( G \) so \( B \) aligns with \( H \).
- If necessary, reflect the triangle to get \( C \) to align with \( E \).
This sequence demonstrates that the two triangles are congruent through rigid motions in the coordinate plane.