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.
Can you make this smaller for me?
To prove the congruence between ΔABC and ΔGHE, we can use a sequence of transformations to show that they are identical in terms of shape and size.
1. Translation: We can move each point of ΔABC 3 units to the left and 4 units up to match the position of ΔGHE. This is done by adding (-3, 4) to the coordinates of ΔABC.
2. Reflection: We can reflect the translated triangle, ΔABC', over the line y = 2 by replacing the y-coordinate of each point with its mirror image with respect to the line. The x-coordinate remains the same.
3. Rotation: We can rotate the reflected triangle, 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).
By performing these transformations, we have transformed ΔABC to ΔGHE, proving their congruence.