Question

Rotate a point on a line at (−4, 6)
180°
counterclockwise. What is the resulting point?(1 point)
Responses

(6, −4)
left parenthesis 6 comma negative 4 right parenthesis

(4, −6)
left parenthesis 4 comma negative 6 right parenthesis

(−4, −6)
left parenthesis negative 4 comma negative 6 right parenthesis

(−6, 4)

Answer 1

To rotate a point 180° counterclockwise around another point, you can use the following formula for the point \( (x, y) \) being rotated around point \( (h, k) \):

1. Subtract the coordinates of the center of rotation from the coordinates of the point:
\[
(x - h, y - k)
\]

2. Rotate by 180° (which is equivalent to negating both coordinates):
\[
(- (x - h), - (y - k))
\]

3. 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 \( (h, k) = (-4, 6) \):

1. Setting \( (x, y) = (-4, 6) \), we find:
\[
(x - h, y - k) = (-4 - (-4), 6 - 6) = (0, 0)
\]

2. Now, we apply the negation:
\[
(-0, -0) = (0, 0)
\]

3. 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:

1. The point to rotate, say \( P = (-4, 6) \) is already the center.
2. 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°.