Rotate a point on a line at (−4, 6) 180° counterclockwise. What is the resulting point?
To rotate a point 180° counterclockwise on a Cartesian plane, we can use the following rotation formula:
(x', y') = (x * cos(θ) - y * sin(θ), x * sin(θ) + y * cos(θ))
Where (x, y) is the original point, (x', y') is the resulting point, and θ is the angle of rotation in radians.
In this case, the original point is (-4, 6), and we want to rotate it 180° counterclockwise.
Converting 180° to radians:
180° * π / 180 = π radians
Substituting the values into the rotation formula:
(x', y') = (-4 * cos(π) - 6 * sin(π), -4 * sin(π) + 6 * cos(π))
Since sin(π) = 0 and cos(π) = -1:
(x', y') = (-4 * (-1) - 6 * 0, -4 * 0 + 6 * (-1))
Simplifying:
(x', y') = (4, -6)
Therefore, the resulting point after rotating (-4, 6) 180° counterclockwise is (4, -6).