Rotation in the Coordinate Plane

Rotation in the Coordinate Plane — Moosa Academy

Turning a point about the origin looks harder than reflecting it, but it needs no trigonometry at all. For quarter turns there are just three rules, and each is a swap, a sign change, or both.

Theorem The three rules

All rotations here are anticlockwise about the origin $O(0,\, 0)$ .

90°:

(x,\, y) \rightarrow (-y,\, x)

— swap, then negate the new first coordinate

180°:

(x,\, y) \rightarrow (-x,\, -y)

— no swap, negate both

270°:

(x,\, y) \rightarrow (y,\, -x)

— swap, then negate the new second coordinate

The 90° and 270° rules are mirror images of each other: both swap, but the minus sign moves to the other slot. The 180° rule is the odd one out — it does not swap at all.

Concept Watching a quarter turn

P(4,\, 2)

turns a quarter circle anticlockwise and lands at

P'(-2,\, 4)

. Applying the rule: swap to get

(2,\, 4)

, then negate the first to get

(-2,\, 4)

Both points sit the same distance from the origin — the turn moves the point around a circle, so its distance from the centre never changes.

Example One point, three turns

Rotate $P(3,\, 2)$ by each of the three angles.

90°: swap to

(2,\, 3)

, negate the first →

(-2,\, 3)

180°: negate both →

(-3,\, -2)

270°: swap to

(2,\, 3)

, negate the second →

(2,\, -3)

Notice the images visit each quadrant in turn: starting in the first, they move to the second, the third, and the fourth.

Note Clockwise turns and full turns

A clockwise turn is just an anticlockwise turn the other way round the circle:

90° clockwise = 270° anticlockwise

270° clockwise = 90° anticlockwise

180° either way gives the same result

A full 360° turn returns every point to where it started, so it changes nothing. The origin itself never moves under any rotation — it is the only fixed point.

Note The rules side by side

Angle	Rule	P(3, 2) becomes
90°	(x, y) → (−y, x)	(−2, 3)
180°	(x, y) → (−x, −y)	(−3, −2)
270°	(x, y) → (y, −x)	(2, −3)

Rotations, like reflections, preserve every length and angle. The image is always congruent to the original — only its position on the plane has changed.

Summary

90° anticlockwise: (x, y) → (−y, x).
180°: (x, y) → (−x, −y).
270° anticlockwise: (x, y) → (y, −x).
The 90° and 270° rules both swap; only the position of the minus differs.
A clockwise turn of 90° equals an anticlockwise turn of 270°.
Distance from the origin is unchanged, so the image is congruent to the original.