To determine the rotation that transforms the point \( P(4, 2) \) to \( P'(-2, 4) \), we can use the following principles of rotation around the origin.
- A 90-degree clockwise rotation of a point \( (x, y) \) results in the new coordinates \( (y, -x) \).
- A 90-degree counterclockwise rotation results in the new coordinates \( (-y, x) \).
- A 180-degree rotation results in the new coordinates \( (-x, -y) \).
Now let's check each possible answer by rotating point \( P(4, 2) \).
Option a: -90 degree rotation (clockwise)
For a -90 degree rotation: \[ P' = (y, -x) = (2, -4) \] This does not match \( P'(-2, 4) \).
Option b: -270 degree rotation (clockwise)
A -270 degree rotation is equivalent to a 90 degree counterclockwise rotation: \[ P' = (-y, x) = (-2, 4) \] This matches \( P'(-2, 4) \).
Option c: 180 degree rotation
For a 180 degree rotation: \[ P' = (-x, -y) = (-4, -2) \] This does not match \( P'(-2, 4) \).
Option d: 90 degree rotation (counterclockwise)
For a 90 degree counterclockwise rotation: \[ P' = (-y, x) = (-2, 4) \] This matches \( P'(-2, 4) \), but since this is interpreted as a positive rotation, it does not satisfy the criteria of rotation given in option b.
Based on the checks, the correct answer is b. -270 degree rotation (clockwise), or equivalently, this also indicates a 90-degree counterclockwise rotation as described in d.
Thus, both b and d describe the same transformation but in different terms. In context, the most appropriate answer according to rotation conventions is one of degree -270 (clockwise).
1 answer