Given a point (-6,4) in a line segment, what is the new point when the line segment is rotated counterclockwise about the origin 180°?
To rotate a point counterclockwise about the origin, we can use the following rotation formula:
(x', y') = (x * cos θ - y * sin θ, x * sin θ + y * cos θ),
where (x', y') are the coordinates of the new point after rotation, (x, y) are the coordinates of the original point, and θ is the angle of rotation.
In this case, the original point is (-6, 4) and we want to rotate it counterclockwise about the origin by 180°.
Using the rotation formula with θ = 180° (or θ = π in radians), we have:
(x', y') = (-6 * cos 180° - 4 * sin 180°, -6 * sin 180° + 4 * cos 180°).
cos 180° = -1 and sin 180° = 0, so the formula becomes:
(x', y') = (-6 * (-1) - 4 * 0, -6 * 0 + 4 * (-1)),
(x', y') = (6, -4).
Therefore, the new point after rotating the line segment counterclockwise about the origin 180° is (6, -4).