Final answer:
To perform a 180-degree rotation of a triangle around a point, find the inverted distance from the point to each vertex, then adjust the coordinates. The provided answer options contain errors as none matches the correct rotated coordinates.
Step-by-step explanation:
To rotate the triangle A(3, 2) B(5, 2) C(4, 4) 180 degrees around the point (-2, 2), you can apply a rotation transformation to each vertex. A 180-degree rotation around a point effectively flips the points over that center point. The transformed points A', B', and C' will be the same distance from the center of rotation as A, B, and C, but in the opposite direction.
To calculate the coordinates after rotation:
- Subtract the center of rotation coordinates from the point's coordinates to translate to the origin.
- Multiply by -1 because a 180-degree rotation is equivalent to reflecting over the origin.
- Add the center of rotation coordinates back to the reflected point's coordinates.
Following these steps:
- A' = [(-1)(3 - (-2)) + (-2), (-1)(2 - 2) + 2] = [(-1)(5) + (-2), 0 + 2] = [-7, 2]
- B' = [(-1)(5 - (-2)) + (-2), (-1)(2 - 2) + 2] = [(-1)(7) + (-2), 0 + 2] = [-9, 2]
- C' = [(-1)(4 - (-2)) + (-2), (-1)(4 - 2) + 2] = [(-1)(6) + (-2), (-1)(2) + 2] = [-8, 0]
The correct answer after fixing the typos would therefore be Triangle A'(-7, 2) B'(-9, 2) C'(-8, 0), but none of the provided options matches these corrected coordinates. Hence, it seems there is a mistake in the listed answer choices.