Question

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 degrees?

Answer 1

To rotate a point counterclockwise about the origin, we can use the formula:

(x', y') = (x * cos(theta) - y * sin(theta), x * sin(theta) + y * cos(theta))

In this case, since we want to rotate the point (−6, 4) 180 degrees counterclockwise, we need to plug in theta = 180 degrees.

Let's calculate the new coordinates:

(x', y') = (-6 * cos(180) - 4 * sin(180), -6 * sin(180) + 4 * cos(180))

Using the fact that cos(180) = -1 and sin(180) = 0, we get:

(x', y') = (-6 * (-1) - 4 * 0, -6 * 0 + 4 * (-1))

Simplifying:

(x', y') = (6, -4)

Therefore, the new point after rotating counterclockwise about the origin 180 degrees is (6, -4).