Symmetry (including Reflection and Rotation)
188. The 90° Spin · swapping coordinates with a sign flip
OP is always perpendicular to OP', and OP = OP'.
90° anticlockwise rotation about O: P(x, y) ↦ P′(−y, x). OP is perpendicular to OP′ (∠POP′ = 90°), and OP = OP′ (distance preserved). Rotation always preserves distances and angles.
What this lesson covers
Try to break it
Drag P anywhere. P' always sits 90° anticlockwise from P, at the same distance from O. If P = (x, y), then P' = (−y, x) — the coordinates swap, and the new x-coordinate flips sign. Try to land P' anywhere else; impossible.
The proof, step by step
Prove that a 90° anticlockwise rotation keeps OP = OP prime with OP perpendicular to OP prime.
- Drop perpendiculars PM and P'N to the x-axis.
- In △OMP and △ONP', ∠OMP = ∠ONP' = 90°.
- OP = OP' (by construction) and ∠MOP = ∠NOP' (both are 90° - ∠MOP').
- △OMP ≅ △ONP' by AAS congruence.
- Therefore, OM = ON and PM = NP'. Since P' is in the second quadrant, its coordinates are (-y, x).
Worked example
A point P(4, 7) is rotated 90° anticlockwise about the origin. What are the coordinates of its image P'?
Applying the rule P(x, y) → P'(-y, x), we substitute x=4 and y=7 to get P'(-7, 4).
- (7, -4)
- (-7, 4) — correct
- (-4, 7)
- (4, -7)