To rotate a point \( (x, y) \) around another point \( (h, k) \) by \( 180^\circ \) counterclockwise, you can use the formula:
\[ (x', y') = (h - (x - h), k - (y - k)) \]
For a \( 180^\circ \) rotation, the new coordinates become:
\[ (x', y') = (2h - x, 2k - y) \]
In this case, the point to rotate is \( (−4, 6) \) and the center of rotation is also \( (−4, 6) \).
Let's substitute:
- \( h = -4 \)
- \( k = 6 \)
- \( x = -4 \)
- \( y = 6 \)
Using the formula:
\[ x' = 2(-4) - (-4) = -8 + 4 = -4 \] \[ y' = 2(6) - 6 = 12 - 6 = 6 \]
So the new point after rotating \( (−4, 6) \) by \( 180^\circ \) around the same point \( (−4, 6) \) remains:
\((−4, 6)\)
However, if we interpret the rotation being around the origin by \( 180^\circ \) instead, the point \( (−4, 6) \) would yield:
\[ (x', y') = (-x, -y) = (4, -6) \]
Thus, if it is indeed a rotation around the origin by \( 180^\circ \), the resulting point would be:
\((4, -6)\)
Since your question does not specify whether the point is rotated about itself or the origin, if you're looking for the transformation around the origin, the correct answer is:
\((4, -6)\)