Question

Describe how to find the coordinates of the image of a point after a 270°
rotation.


WRITER

Answer 1

To rotate a point (x,y) by 270° in the plane, first be clear about the direction:

- 270° counterclockwise (CCW) = 90° clockwise (CW).
- 270° clockwise (CW) = 90° counterclockwise (CCW).

For a rotation about the origin:

- 270° CCW (or 90° CW): (x,y) → (y, −x).
- 270° CW (or 90° CCW): (x,y) → (−y, x).

You can get this from the rotation matrix for angle θ:
x' = x cosθ − y sinθ
y' = x sinθ + y cosθ
Plugging θ = 270° (cos270° = 0, sin270° = −1) gives (x',y') = (y, −x).

Example: Rotate (3,2) by 270° CCW → (2, −3).

If rotating about a point (h,k) instead of the origin:
1. Translate so (h,k) is origin: (x−h, y−k).
2. Apply the rotation formula above.
3. Translate back: add (h,k) to the result.