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1. Translation:
We can translate ΔABC to match the position of ΔGHE by moving each point 3 units to the left and 4 units up. This can be done by adding (-3, 4) to each coordinate of ΔABC.
ΔABC' = (-2, 1) (-2, 3) (1, 1)
2. Reflection:
We can reflect ΔABC' over the line y = 2. This can be done by replacing the y-coordinate of each point with its mirror image with respect to the line. The x-coordinate remains the same.
ΔABC'' = (-2, 3) (-2, 1) (1, 3)
3. Rotation:
We can rotate ΔABC'' 180 degrees clockwise around the point (-2, 2). This moves each point to a new position based on the rotation rule: (x', y') = (x*cosθ - y*sinθ, x*sinθ + y*cosθ)
Applying this transformation to each point of ΔABC'', we get:
ΔGHE = (-2, 2) (-4, 2) (-2, 5)
Therefore, by performing a sequence of translation, reflection, and rotation, we have transformed ΔABC to ΔGHE, proving their congruence.
1. ΔABC can be translated by adding (-3, 4) to each coordinate, resulting in ΔABC' = (-2, 1) (-2, 3) (1, 1).
2. ΔABC' can be reflected over y = 2 to obtain ΔABC'' = (-2, 3) (-2, 1) (1, 3).
3. By rotating ΔABC'' 180 degrees clockwise around (-2, 2), we get ΔGHE = (-2, 2) (-4, 2) (-2, 5).
Thus, we have transformed ΔABC to ΔGHE using translation, reflection, and rotation, proving their congruence.