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Invariant Points

HigherHigher tier onlyAQAEdexcelOCR

Practise invariant points with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through identifying invariant points under transformations, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Invariant points don't move when the transformation is applied.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An invariant point is one that does not move when a transformation is applied. Identifying them is a quick way to check a transformation has been carried out correctly.

Different transformations have different invariant points. In a reflection, every point on the mirror line is invariant. In a rotation, only the centre of rotation is invariant. In an enlargement, only the centre of enlargement stays put.

A translation has no invariant points at all, because every point moves by the same vector. That makes it the only one of the four with none.

Revision notes

Invariant points by transformation

Reflection: all points on the mirror line. Rotation: the centre only. Enlargement: the centre only.

Translation: none, since every point moves.

Using invariant points as a check

If a vertex of your shape lies on the mirror line, it should appear in the same place on the image.

If it has moved, the reflection has been drawn incorrectly.

Identifying invariant points in a question

Look for points shared between the object and image.

If a whole edge is unchanged, the transformation is likely a reflection in the line containing that edge.

Key points

  • An invariant point does not move.
  • All points on a mirror line are invariant.
  • Only the centre is invariant in a rotation.
  • Only the centre is invariant in an enlargement.
  • A translation has no invariant points.
  • Use them to check your transformation.

Worked examples

Example 1

How many invariant points does a translation have?

Working

\[\text{Every point moves by the same vector}\]no point stays still
\[0\]state the answer

Example 2

A shape is reflected in the line \(y = 2\). Which points are invariant?

Working

\[\text{Points on the mirror line do not move}\]apply the rule for reflections
\[\text{All points on } y = 2\]state the invariant points

Example 3

A shape is rotated about \((1, 3)\). Which point is invariant?

Working

\[\text{Only the centre stays fixed}\]apply the rule for rotations
\[(1, 3)\]state the invariant point

Common mistakes

  • Thinking a rotation leaves several points fixed.

    Only the centre of rotation is invariant.

  • Saying a translation has invariant points.

    Every point moves by the same vector, so there are none.

  • Confusing invariant points with congruence.

    A shape can be congruent to its image while having no invariant points.

  • Missing that a whole line can be invariant.

    In a reflection, every point on the mirror line stays put.

Exam tips

  • Learn which transformation leaves which points fixed.
  • Use invariant points to check your drawing.
  • Look for shared points between object and image.
  • Remember a translation is the only one with none.

Key terms

Invariant point
A point unchanged by a transformation.
Mirror line
The line of reflection, entirely invariant.
Centre of rotation
The single invariant point in a rotation.
Vector
The description of a translation.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.