Math101 · Independent Ontario learning library
Math101learn.math101.caRotations
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.
Worked example
Representations and interpretation
A compass arc centred at $C$ keeps radius fixed while angle marks the turn. Coordinate matrices encode the same action; for $90^\circ$ CCW, $\begin{pmatrix}0&-1\\1&0\end{pmatrix}(x,y)=(-y,x)$.
Reasoning about variations
A $270^\circ$ counterclockwise turn equals $90^\circ$ clockwise. Two reflections across intersecting lines compose to a rotation by twice the angle between the lines.
Common mistakes
How to check your work
- Compare distances from centre before and after.
- Compute the directed angle between position vectors.
- Apply the inverse rotation and recover the original point.
Practice
- Rotate $(3,-4)$ $90^\circ$ counterclockwise about the origin.
- Rotate $(-2,5)$ $180^\circ$ about the origin.
- Does a rotation reverse orientation?
Answers and brief solutions
Show answers
- $(4,3)$ Use $(x,y)\mapsto(-y,x)$.
- $(2,-5)$ Both coordinates change sign.
- No A rotation is an orientation-preserving rigid motion.
Synthesis and transfer
A quarter-turn of a design about a non-origin centre is safest after translating the centre to the origin, applying the coordinate rule, and translating back.
To rotate $P=(5,3)$ by $90^\circ$ counterclockwise around $C=(2,1)$, first use relative vector $(3,2)$. The quarter-turn sends it to $(-2,3)$, and adding the centre back gives image $(0,4)$. Distances $CP$ and $CP'$ are both $\sqrt{13}$, while the directed angle between the vectors is $90^\circ$. Omitting either translation would rotate around the origin instead. The inverse is a $90^\circ$ clockwise turn about the same centre, and applying it to the image should recover the original coordinates.
Related topics
Teaching and accessibility note
Explore the idea
Vector and matrix transform
Change one quantity at a time and connect what moves to Rotations.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Rotate $(3,-4)$ $90^\circ$ counterclockwise about the origin.
- Use $(x,y)\mapsto(-y,x)$.
End of lesson
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