Asked by con
△ABC is congruent to △GHE. Describe a sequence of transformations from △ABC to △GHE that proves their congruence.
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GPT-5 mini
AI
We want an isometry (rigid motion) sending A→G, B→H, C→E. One convenient 3-step sequence that always works:
1. Translate ΔABC by the vector GA (i.e. slide so A lands on G). Now A is at G; call the moved triangle A'B'C' with A' = G.
2. Rotate the translated triangle about point G through the angle that carries ray GA'G B' (the direction of A'→B') onto the direction of G→H. Choose the rotation direction so that B' lands on H. After this rotation B' = H.
3. If after the rotation the image of C' is not E, reflect the rotated triangle across line GH. (If the rotation already sent C' to E, no reflection is needed.)
After these steps the image of ΔABC is exactly ΔGHE, proving the triangles are congruent (translations, rotations and reflections preserve distance and angle).
1. Translate ΔABC by the vector GA (i.e. slide so A lands on G). Now A is at G; call the moved triangle A'B'C' with A' = G.
2. Rotate the translated triangle about point G through the angle that carries ray GA'G B' (the direction of A'→B') onto the direction of G→H. Choose the rotation direction so that B' lands on H. After this rotation B' = H.
3. If after the rotation the image of C' is not E, reflect the rotated triangle across line GH. (If the rotation already sent C' to E, no reflection is needed.)
After these steps the image of ΔABC is exactly ΔGHE, proving the triangles are congruent (translations, rotations and reflections preserve distance and angle).
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