GeoProof — Rotation about a Point explorer

every point keeps its distance from O and turns through the same angle

Drag the angle handle around O to spin the image. Distances OA = OA′ are preserved; orientation is kept.

Rotation about a Point

A rotation turns a figure about a fixed centre O by a chosen angle. Each point P moves to P′ so that:

OP′ = OP and ∠POP′ = the rotation angle

So every point travels along a circular arc centred at O, and they all sweep through the same angle.

What is preserved

Rotation is a rigid motion (isometry): lengths and angles are unchanged, so the image is congruent to the original. Unlike a reflection, a rotation keeps orientation — ABC and A′B′C′ run the same way around.

The centre is fixed

Only the centre O stays put; every other point moves (unless the angle is 0° or 360°). Put O at a vertex and that vertex's image stays glued to it.

Rotational symmetry

A shape has rotational symmetry if some rotation maps it onto itself — a square returns to itself every 90°, an equilateral triangle every 120°.

Two reflections = a rotation

Reflecting across two lines that meet at O is the same as a single rotation about O, through twice the angle between the lines.