To demonstrate that triangles \( \triangle ABC \) and \( \triangle GHE \) are congruent, we can describe a sequence of transformations that takes \( \triangle ABC \) to \( \triangle GHE \).
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Translation: First, we can translate \( \triangle ABC \) so that point A moves to point G. The coordinates of A are \((1, -3)\) and the coordinates of G are \((-2, 2)\). To find the translation vector, we calculate: \[ \text{Translation vector} = G - A = (-2, 2) - (1, -3) = (-3, 5) \] So, we translate \( \triangle ABC \) by the vector \((-3, 5)\).
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Rotation: After the translation, we need to rotate \( \triangle ABC' \) (the translated triangle) about point G to match triangle GHE. To confirm the correct angle and direction, we can analyze the orientation of the triangles and the angles formed. In this case, you may find the angle between line segments in \( \triangle ABC' \) and the corresponding segments in \( \triangle GHE \).
For instance, if triangle \( \triangle ABC' \) has a particular orientation, you would want to rotate it around point G to align it perfectly with \( \triangle GHE \). Assuming a counterclockwise rotation to match angles may often require knowledge of specific angles or distances which can be calculated based on the corresponding vertices.
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Final Position: After applying the translation and the appropriate rotation, we should have \( \triangle ABC \) aligned perfectly with \( \triangle GHE \). You might also need to reflect if necessary based on the orientation to further ensure both triangles overlap perfectly.
Thus, the sequence of transformations can be summarized as:
- Translate \( \triangle ABC \) by the vector \((-3, 5)\) to obtain the new triangle \( A'B'C' \).
- Rotate \( A'B'C' \) around point G to match the orientation of triangle \( GHE \).
This sequence of transformations proves that \( \triangle ABC \) is congruent to \( \triangle GHE \).