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 rewrite this for me?
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.