Rotation in the plane: center, angle, sense, image of a point

The 3 elements

A rotation needs:

1. Center $O$ (fixed point)
2. Angle $\theta$ (degrees or radians)
3. Sense:
   • Counter-clockwise (positive, trigonometric)
   • Clockwise (negative)

Image of point M

The image $M'$ satisfies:

• $OM = OM'$ (same distance from center)
• $\widehat{MOM'} = \theta$ (angle from $M$ to $M'$, correct sense)

Step-by-step construction

1. Draw the circle centered at $O$ with radius $OM$.
2. Draw the ray $[OM)$.
3. Draw another ray from $O$ making angle $\theta$ with $[OM)$ (correct sense).
4. The intersection with the circle is $M'$.

Properties

A rotation preserves:

• Lengths
• Angles
• Areas
• Orientation (unlike reflection)

Special cases

• 0° or 360°: identity.
• 180°: central symmetry with center $O$.
• 90°: squares, 4-leaf shapes.

Common mistakes

1. Forgetting sense (CW vs CCW).
2. Confusing rotation and translation: rotation has fixed center; translation slides everything.
3. Thinking the center moves: it’s always invariant.

?À toi de jouer

A rotation by 180° around O equals:

A.Central symmetry with center OB.The identityC.An axial reflectionD.A translation

Valider