To rotate a point 180° counterclockwise around another point, you can use the following formula for the point \( (x, y) \) being rotated around point \( (h, k) \):
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Subtract the coordinates of the center of rotation from the coordinates of the point: \[ (x - h, y - k) \]
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Rotate by 180° (which is equivalent to negating both coordinates): \[ (- (x - h), - (y - k)) \]
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Add back the coordinates of the center of rotation: \[ (h - (x - h), k - (y - k)) \]
Let's apply this to the point \( (-4, 6) \) rotating around itself \( (h, k) = (-4, 6) \):
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Setting \( (x, y) = (-4, 6) \), we find: \[ (x - h, y - k) = (-4 - (-4), 6 - 6) = (0, 0) \]
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Now, we apply the negation: \[ (-0, -0) = (0, 0) \]
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Finally, we add back the center of rotation: \[ (-4 + 0, 6 + 0) = (-4, 6) \]
Since we are rotating the point \( (-4, 6) \) itself, it remains unchanged.
If you meant to rotate a different point, please clarify. However, if we consider the point directly opposite to \( (-4, 6) \) across the center of rotation, the process would be:
- The point to rotate, say \( P = (-4, 6) \) is already the center.
- Therefore the correct point after a 180° rotation around \( (-4, 6) \) does not change.
If, however, you were rotating the point \( (0, 0) \) around \( (-4, 6) \), the result would be: Initial relative position: \[ (0 - (-4), 0 - 6) = (4, -6) \] Negated: \[ (-4, 6) \] Finally, adding the center back: \[ (-4 - 4, 6 - 6) = (-8, 0) \] This doesn't seem to be the requested point based on the options provided.
From the answers given, the closest matchup logically would be retracing to a coordinate that reflects moving through the x and y axis, leading us to interpret:
Reflectively the point across original axes leading to the final status toward the upper answer. The point reflecting across these axes from (-4, 6) would deliver:
The point of interest remains within rotations reflecting consistently backward by 180° would thus yield
Consequently the answer appropriate: as accurate as the original endpoints would serve their original bounds.
To finalize: \[ \text{None of the listed points (except itself) correctly display what is re-rotated in 180 degrees.} \]
Final Answer: None of the available options depict the inherently rotated 180°.