Math101 · Independent Ontario learning library
Math101Rotations
A rotation turns every point through the same directed angle about a fixed centre. Distance from the centre and all shape measurements are preserved, so rotations are rigid transformations and preserve orientation.
Rotations describe symmetry, gears, graphics, navigation, and congruence. Centre-relative thinking generalizes simple origin rules.
Intuition and core definition
A rotation turns every point through the same directed angle about a fixed centre. Distance from the centre and all shape measurements are preserved, so rotations are rigid transformations and preserve orientation. The centre is the only fixed point for a non-full-turn rotation.
Notation, language, and conditions
Positive angles conventionally rotate counterclockwise. About the origin: $90^\circ$ counterclockwise maps $(x,y)$ to $(-y,x)$; $180^\circ$ maps to $(-x,-y)$; $270^\circ$ counterclockwise maps to $(y,-x)$. Other centres require translating relative coordinates first.
Why this idea matters
A rotation preserves shape and orientation around a fixed centre while changing each point's direction by one common angle.
A dependable method
- Identify centre, angle magnitude, and direction.
- Draw or calculate each point’s vector from the centre.
- Rotate that vector using geometry or a coordinate rule.
- Add the centre coordinates back if the centre is not the origin.
- Check equal centre distances, angle turn, orientation, and side lengths.
