Rotations

What Is a Rotation?

A rotation turns a shape around a fixed point called the centre of rotation.

The size of the shape stays exactly the same.

Only its position and orientation change.

If a point lies on the centre of rotation, it doesn’t move. We call this an invariant point.

Key Facts

• Rotations are always given in degrees — usually $90\degree$, $180\degree$, or $270\degree$.
• Rotations can be clockwise or anticlockwise.
• $90\degree$ clockwise = $270\degree$ anticlockwise (they end in the same place)
• A rotation is a type of congruent transformation — the image is the same shape and size as the object.

How to Rotate a Shape

Step-by-Step:

Step 1: Place tracing paper over the shape and draw around it.

Step 2: Mark the centre of rotation on the tracing paper.

Step 3: Draw an arrow facing upwards (helps track rotation).

Step 4: Put your pencil on the centre and turn the tracing paper the required angle in the given direction.

Step 5: Copy the new shape onto your grid and label it.

Example

Rotate a shape $90\degree$ clockwise

Question: Rotate triangle A $90\degree$ clockwise about the point (1, 1). Label your new shape A'.

Solution:

• Draw triangle A on tracing paper.
• Mark the centre of rotation (1, 1).
• Place your pencil on the centre, and rotate the tracing paper $90\degree$ clockwise.
• Copy the rotated shape onto the grid.

Triangle A was originally at points:

$$A = (2, 2),\quad B = (3, 2),\quad C = (3, 3)$$

After rotation, the new coordinates are:

$$A' = (1, 0),\quad B' = (1, -1),\quad C' = (2, -1)$$

Plot and label the new triangle.

Example

Describing a Rotation

Question: Shape A has been transformed into shape B. Describe fully the single transformation.

Solution:

• The shape is the same size and same orientation, but its position has changed.
• The image appears to be rotated.

Try using tracing paper:

• Draw shape A on the paper.
• Try rotating it $90\degree$ anticlockwise about the point (0, -2).
• When it matches shape B, that’s your transformation

So, the transformation is:

Rotation, $90\degree$ anticlockwise, centre (0, -2)

Reversing a Rotation

To reverse a rotation:

• Keep the same centre
• Use the same angle
• Change the direction (clockwise ↔ anticlockwise)

Example:

• A shape is rotated $90\degree$ clockwise about (3, 1)
• To reverse: rotate $90\degree$ anticlockwise about (3, 1)