To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle GHE \) through transformations, we can use a sequence of rigid motions. Rigid motions preserve distances and angles, which are essential for congruence. Here’s a general approach you can take for your proof:
-
Translation: Start by translating \( \triangle ABC \) so that one of its vertices, say vertex \( A \), coincides with vertex \( G \). This means you move the entire triangle along a straight path without changing its shape or size.
-
Rotation: After translating \( \triangle ABC \) to position \( A \) at \( G \), the next step is to rotate \( \triangle ABC \) about point \( G \) (the new position of vertex \( A \)) so that vertex \( B \) aligns with vertex \( H \). The angle of rotation will depend on the relative positions of points \( B \) and \( H \).
-
Reflection (if necessary): If, after rotation, the triangle \( ABC \) does not match \( \triangle GHE \) exactly (which could happen if the triangles are mirror images), you may need to perform a reflection across a line (for example, the line through points \( G \) and \( H \)).
After completing these steps, triangle \( \triangle ABC \) will coincide perfectly with triangle \( \triangle GHE \). This sequence of transformations (translation, rotation, and possibly reflection) shows that \( \triangle ABC \) is congruent to \( \triangle GHE \).
To summarize:
- Translate \( \triangle ABC \) so \( A \) coincides with \( G \).
- Rotate the triangle around point \( G \) until \( B \) coincides with \( H \).
- If necessary, reflect \( \triangle ABC \) across the appropriate line to achieve complete overlap with \( \triangle GHE \).
This sequence of transformations proves the congruence of the two triangles.