To prove the congruence between ΔABC and ΔGHE, we need to show that they are identical in terms of shape and size.

Question

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 rewrite this for me?

Answer 1

To prove that ΔABC is congruent to ΔGHE, we need to demonstrate that both triangles have the same shape and size. This can be shown through a series of transformations:

1. Translation:
We can translate ΔABC by moving each point 3 units to the left and 4 units up. By adding (-3, 4) to the coordinates of each point in ΔABC, we obtain ΔABC' = (-2, 1) (-2, 3) (1, 1).

2. Reflection:
Next, we reflect ΔABC' over the line y = 2. To achieve this, we replace the y-coordinate of each point with its mirror image with respect to the line while keeping the x-coordinate unchanged. This yields ΔABC'' = (-2, 3) (-2, 1) (1, 3).

3. Rotation:
To complete the congruence transformation, we rotate ΔABC'' 180 degrees clockwise around the point (-2, 2). This will reposition each point based on the rotation rule: (x', y') = (x*cosθ - y*sinθ, x*sinθ + y*cosθ). Applying this rule to the points of ΔABC'', we obtain ΔGHE = (-2, 2) (-4, 2) (-2, 5).

By performing a sequence of translation, reflection, and rotation, we have successfully transformed ΔABC into ΔGHE, proving their congruence.