To rotate a point \(180^\circ\) counterclockwise around another point, you effectively invert the coordinates of the point relative to the center of rotation.
In this case, the point to be rotated is \((-4, 6)\), and we are rotating it around the point \((-4, 6)\) itself.
When a point is rotated \(180^\circ\) around itself, the point remains unchanged. Therefore, the resulting point after the rotation is:
\[ (-4, 6) \]
However, since the question seems to imply that we might be looking for the coordinates resulting from a rotation around the origin or some point not specified as the center, let's clarify with respect to straightforward rotation around the origin:
To rotate \((-4, 6)\) around the origin \(180^\circ\), you use the formula: \[ (x', y') = (-x, -y) \] Applying this: \[ x' = -(-4) = 4 \] \[ y' = -6 \] Thus, the rotated point around the origin \(180°\) counterclockwise from \((-4, 6)\) becomes \((4, -6)\).
Among the provided choices, the correct answer is: (4, -6).