Rotate a point on a line at (−4, 6)

Question

180°
counterclockwise. What is the resulting point?(1 point)
Responses

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

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

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

(6, −4)

Answer 1

To rotate a point \( (x, y) \) by \( 180^\circ \) counterclockwise about the origin, you can use the transformation:

\[ (x, y) \to (-x, -y) \]

Now, we first move the given point \( (-4, 6) \) to the origin by translating it. However, since we are rotating about the point itself, we can directly apply the transformation as if it were about the origin:

1. Translate the point to the origin: This step is conceptually helpful but, in this case, we can just apply the transformation since we are rotating the point about itself.

2. Apply the rotation: \[ (-4, 6) \to (-(−4), −6) = (4, −6) \]

Therefore, the resulting point after rotating \( (-4, 6) \) by \( 180^\circ \) counterclockwise is:

\[ (4, -6) \]

Thus, the answer is:

(4, -6)