△ABC is congruent to △GHE . Describe a sequence of transformations from △ABC to △GHE that proves their congruence.
ABC = (1,-3) (1,-1) (4,-3)
GHE = (-2,2) (-4,2) (-2,5)
To prove the congruence between ΔABC and ΔGHE, we need to show that they are identical in terms of shape and size.
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.