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 \( (x, y) \) 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:

\((4, -6)\)