Question

Given a point on a line segment (-4,5), what is the new point when the point is rotated counterclockwise about the origin, 270°?

Answer 1

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

x_new = x*cos(theta) - y*sin(theta)
y_new = x*sin(theta) + y*cos(theta)

In this case, the point to be rotated is (-4, 5) and the rotation angle is 270°.

First, we need to convert the angle from degrees to radians, since the trigonometric functions in the rotation formula work with radians.

270° = (270/180)π = (3/2)π

Now, applying the rotation formula:

x_new = (-4)*cos((3/2)π) - 5*sin((3/2)π)
= (-4)*cos(-(1/2)π) - 5*sin(-(1/2)π)
= (-4)*0 - 5*(-1)
= 0 + 5
= 5

y_new = (-4)*sin((3/2)π) + 5*cos((3/2)π)
= (-4)*sin(-(1/2)π) + 5*cos(-(1/2)π)
= (-4)*(-1) + 5*0
= 4 + 0
= 4

Therefore, the new point after rotating (-4, 5) counterclockwise about the origin by an angle of 270° is (5, 4).