Rotate a point on a line at (-4,6) 180 degrees counterclockwise. What is the resulting point
To rotate a point 180 degrees counterclockwise, we need to reflect it over the x-axis, then reflect it over the y-axis.
First, reflecting over the x-axis will change the y-coordinate to its opposite:
(-4,6) --> (-4,-6)
Next, reflecting over the y-axis will change the x-coordinate to its opposite:
(-4,-6) --> (4,-6)
Therefore, the resulting point is (4,-6).
To rotate a point 180 degrees counterclockwise, we need to apply a rotation formula. The formula for a counterclockwise rotation around the origin is:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
Since we want to rotate the point (-4,6) 180 degrees counterclockwise, we need to use theta = 180 degrees.
Converting theta to radians:
180 degrees = pi radians
Substituting the values in the rotation formula:
x' = -4 * cos(pi) - 6 * sin(pi)
y' = -4 * sin(pi) + 6 * cos(pi)
Calculating the sine and cosine of pi:
cos(pi) = -1
sin(pi) = 0
Substituting in the values:
x' = -4 * (-1) - 6 * 0
= 4 - 0
= 4
y' = -4 * 0 + 6 * (-1)
= 0 - 6
= -6
Therefore, the resulting point after rotating (-4,6) 180 degrees counterclockwise is (4,-6).