Describe the changes and invariance achieved by combinations of rotations,…

The idea

A combination means performing two or more transformations in sequence on a shape. You must state what changes and what stays the same. Rotations turn a shape about a fixed centre by a given angle and direction; reflections flip it across a mirror line; translations slide it by a column vector. Invariance means properties unchanged by the combination. For example, reflect a triangle in the x-axis then rotate 90° anticlockwise about the origin: the final image has the same area, side lengths and angles as the original, but its orientation and position change. A single reflection reverses orientation; two reflections in parallel lines give a translation; two reflections in intersecting lines give a rotation. Describing the combination means naming the overall single transformation that maps the original object to the final image, or listing the changes and invariants precisely.

What earns the marks

• Identify the individual transformations in the combination and the order in which they are applied.
• State that lengths, angles and area are invariant under any combination of rotations, reflections and translations.
• Recognise that orientation is reversed by an odd number of reflections and preserved by an even number.
• Determine the single equivalent transformation: two parallel reflections give a translation; two intersecting reflections give a rotation about the intersection point.
• Describe the overall change in position and orientation of the image compared with the object.

Common mistakes and how to put them right

• Thinking that a combination of transformations changes the size or shape of the figure.

  Put it right: Rotations, reflections and translations are all congruences, so lengths, angles and area are invariant.

• Assuming the order of transformations never matters.

  Put it right: In general the final image depends on the order, so apply them in the stated sequence.

• Believing that two reflections always produce a reflection.

  Put it right: Two reflections in parallel lines produce a translation, and two reflections in intersecting lines produce a rotation.

• Confusing orientation with position.

  Put it right: Orientation describes the order of vertices (clockwise or anticlockwise), while position is where the shape sits on the grid.

Exam tips

• Trace the shape through each transformation in turn, labelling the intermediate image, so the final position is clear.
• Check whether the number of reflections is odd or even to decide whether orientation is reversed or preserved.
• When asked for the single equivalent transformation, identify its type first, then give the centre, angle and direction for a rotation or the column vector for a translation.