Geometry and measures - Properties and constructions: identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional and negative scale factors) (Higher tier)

The idea

Transformations move or resize shapes on a coordinate grid. A translation slides a shape without turning it, described by a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$. A reflection flips a shape across a mirror line, such as the x-axis or the line `y = x`. A rotation turns a shape about a centre of rotation by a given angle and direction (clockwise or anticlockwise). An enlargement changes size from a centre of enlargement by a scale factor. A fractional scale factor such as ½ makes the image smaller; a negative scale factor places the image on the opposite side of the centre and inverts it. For example, scale factor -2 doubles side lengths and flips the image through the centre. Congruent shapes are identical in shape and size and can be mapped by rotation, reflection or translation. Similar shapes have equal angles and proportional sides, related by an enlargement.

What earns the marks

• Correctly identify the single transformation (rotation, reflection, translation or enlargement) that maps one shape to another.
• Give a full description: for a rotation state centre, angle and direction; for a reflection state the mirror line; for an enlargement state centre and scale factor; for a translation give the column vector.
• Draw the image of a shape after a transformation, including enlargements with fractional or negative scale factors.
• Use a column vector to describe a translation, e.g. $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ moves a shape 3 right and 2 down.
• Explain why a scale factor of -1 gives the same image as a rotation of 180° about the centre, and calculate image coordinates by reversing the step from the centre and scaling it.

Common mistakes and how to put them right

• Confusing a rotation with a translation or a reflection, e.g. describing a 90° turn as a slide.

  Put it right: Check whether the shape turns about a point (rotation), slides (translation) or flips (reflection).

• Forgetting one part of a description, such as the centre of rotation or the equation of the mirror line.

  Put it right: State every required element.

• Enlarging from the origin instead of the specified centre of enlargement.

  Put it right: Measure each vertex's distance from the given centre and multiply by the scale factor.

• Confusing congruent (identical shape and size) with similar (identical shape, different size).

  Put it right: Congruent shapes match exactly; similar shapes have equal angles and proportional sides.

Exam tips

• Use tracing paper to help identify rotations and reflections: rotate the paper around a test point or flip it over a line.
• To find a centre of enlargement, draw straight lines through corresponding points on the object and image; the point where these lines cross is the centre.