Rotation about a Fixed Point

A rotation turns a figure about a fixed point, the centre of rotation, through a given angle. Every point sweeps around the centre while keeping the same distance from it.

Theorem Rotation about a centre

Under a rotation, the distance from the centre never changes, so the image is congruent to the original. The centre itself is the only point that stays put.

To rotate a whole figure, turn each vertex about the centre by the same angle, then join the new vertices.

Theorem The rotation rules about the origin

For counter-clockwise rotations about the origin, the coordinates follow fixed rules:

90°:

(x,y) \to (-y,\ x)

180°:

(x,y) \to (-x,\ -y)

270°:

(x,y) \to (y,\ -x)

360°:

(x,y) \to (x,\ y)

Example Applying the rules

Rotate (2, 1) by 90°: multiply y by −1, then swap ⟹

(2,1) \to (-1,\ 2)

Rotate (5, 2) by 180°: multiply both by −1 ⟹

(5,2) \to (-5,\ -2)

Rotate (5, 2) by 270°: multiply x by −1, then swap ⟹

(5,2) \to (2,\ -5)

⟹ each rule turns the point about the origin without changing its distance.

Note 90° vs 270°

Both rules swap the coordinates; only the sign differs. For 90° you negate the y first, giving $(-y,\,x)$ ; for 270° you negate the x first, giving $(y,\,-x)$ .

Summary

A rotation keeps every point's distance from the centre; only its position changes.
The point at the centre does not move, for any angle.
About the origin: 90° → $(-y,x)$ , 180° → $(-x,-y)$ , 270° → $(y,-x)$ , 360° → $(x,y)$ .
These coordinate rules apply to rotations about the origin (0, 0) only.