Rotations

• Rotation is a transformation where every point of a figure moves along a circular path around a fixed point known as the centre of rotation.

• The angle of rotation determines how far the figure is turned, typically measured in degrees ($90^{\circ}$, $180^{\circ}$, or $270^{\circ}$).

• The direction of rotation specifies whether the turn is clockwise (CW) or anti-clockwise (ACW); note that a $90^{\circ}$ CW rotation is equivalent to a $270^{\circ}$ ACW rotation.

• Rotations are isometries, which means the size and shape of the object are preserved, resulting in an image that is congruent to the original object.

Coordinate Rules (Rotation about Origin)

• $90^{\circ}$ Anti-clockwise / $270^{\circ}$ Clockwise: The coordinates transform as $(x, y) \to (-y, x)$.
• $180^{\circ}$ (Either direction): The coordinates transform as $(x, y) \to (-x, -y)$.
• $270^{\circ}$ Anti-clockwise / $90^{\circ}$ Clockwise: The coordinates transform as $(x, y) \to (y, -x)$.

Practical Execution

• Tracing Paper Method: Trace the object and the centre, place a pencil on the centre, and rotate the paper by the required angle to find the image coordinates.
• Vector Method: For rotations not about the origin, translate the centre to $(0,0)$, apply the coordinate rule, and then translate back.

• Directionality: Rotations require a direction (CW/ACW) unless the angle is $180^{\circ}$, whereas reflections only require a line of symmetry.

• Full Description: To gain full marks when describing a rotation, you must state four things: the word 'Rotation', the centre (as coordinates), the angle, and the direction.

• The $180^{\circ}$ Exception: If the rotation is $180^{\circ}$, you do not need to specify a direction because clockwise and anti-clockwise turns result in the same position.

• Finding the Centre: If the angle is $180^{\circ}$, draw lines connecting corresponding vertices; the point where all these lines intersect is the centre of rotation.

• Sanity Check: Always verify that the distance from a vertex to the centre is identical for both the object and the image.

• Direction Confusion: Students often confuse clockwise and anti-clockwise; remember that 'anti-clockwise' follows the order of quadrants in a standard coordinate system (I to II to III to IV).

• Missing the Centre: A common error is describing the angle and direction but forgetting to provide the coordinates for the centre of rotation.

• Incorrect Origin Assumption: Do not assume the centre is always $(0,0)$; always check the problem description or use tracing paper to verify the pivot point.

• Sign Errors: When using coordinate rules, ensure the signs are applied correctly (e.g., in a $90^{\circ}$ ACW rotation, the new x-coordinate is the negative of the old y-coordinate).