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GCSE/Mathematics/WJEC

G8Changes and invariance under rotation, reflection, translation, enlargement combinations

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Notes

Transformations and what stays the same

WJEC examines four transformations: translation, reflection, rotation and enlargement. Each preserves some properties and changes others.

The four transformations

TypeWhat changesWhat stays the same
TranslationPositionSize, shape, orientation
ReflectionPosition, orientation (left/right flip)Size, shape
RotationPosition, orientationSize, shape
EnlargementPosition, sizeShape (similar), orientation if k > 0

The first three are CONGRUENCE transformations — image and object are congruent. Enlargement is a SIMILARITY transformation (unless k = ±1).

Describing each

Translation — vector (a, b) where a is the horizontal shift and b vertical.

Reflection — line of reflection (e.g. y-axis, x-axis, y = x, y = −x, y = k, x = h).

Rotation — three pieces: angle, direction (clockwise / anticlockwise), centre of rotation.

Enlargement — three pieces: scale factor (with sign), centre of enlargement.

Combinations of transformations

Apply transformations one after another. Order MATTERS — usually a different combined effect.

Worked: rotate 90° anticlockwise about origin, then translate by (3, −2).

Point (4, 1) → after rotation → (−1, 4) → after translation → (2, 2).

Worked: reflect in y = x, then enlarge by scale factor 2 about origin.

Point (3, 5) → reflect → (5, 3) → enlarge → (10, 6).

Invariance

A point or shape is invariant under a transformation if it does NOT move.

  • Translation by non-zero vector: no invariant points.
  • Reflection: every point on the line of reflection is invariant.
  • Rotation about centre O: only O is invariant.
  • Enlargement about centre O: only O is invariant.

WJEC exam tip

When two transformations combine to give a single equivalent transformation, sketch first. The key clue: if the result preserves orientation, it is rotation/translation; if it flips orientation, it involves an odd number of reflections. Stating the equivalent single transformation needs all describing details (e.g. "rotation, 180°, centre (2, 1)") for the A1 marks.

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Practice questions

Try each before peeking at the worked solution.

  1. Question 13 marks

    Identify a single transformation

    WJEC Unit 1 (Non-calculator) — Intermediate

    Triangle T has vertices A(1, 1), B(3, 1), C(1, 4). Triangle T' has vertices A'(−1, −1), B'(−3, −1), C'(−1, −4).

    Describe fully the single transformation that maps T onto T'. (3 marks)

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  2. Question 25 marks

    Combine transformations and find invariant points

    WJEC Unit 2 (Calculator) — Higher

    Triangle P has vertices (2, 3), (5, 3), (5, 7). Triangle P is reflected in the line y = x to give triangle Q. Triangle Q is then translated by the vector (4, −2) to give triangle R.

    (a) Find the coordinates of the vertices of triangle R. (3 marks)
    (b) State whether any point of triangle P is invariant under the combined transformation, justifying your answer. (2 marks)

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  3. Question 34 marks

    Equivalent single transformation

    WJEC Unit 1 (Non-calculator) — Higher

    Shape S is reflected in the x-axis to give shape S₁, then S₁ is reflected in the y-axis to give shape S₂.

    (a) The single transformation that maps S onto S₂ is a rotation. State the angle and centre of rotation. (3 marks)
    (b) Does this single rotation have any invariant points of S? Explain. (1 mark)

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  4. Question 43 marks

    Identify a single transformation

    WJEC Unit 1 (Non-calculator) — Intermediate

    Triangle T has vertices A(1, 1), B(3, 1), C(1, 4). Triangle T' has vertices A'(−1, −1), B'(−3, −1), C'(−1, −4).

    Describe fully the single transformation that maps T onto T'. (3 marks)

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  5. Question 55 marks

    Combine transformations and find invariant points

    WJEC Unit 2 (Calculator) — Higher

    Triangle P has vertices (2, 3), (5, 3), (5, 7). Triangle P is reflected in the line y = x to give triangle Q. Triangle Q is then translated by the vector (4, −2) to give triangle R.

    (a) Find the coordinates of the vertices of triangle R. (3 marks)
    (b) State whether any point of triangle P is invariant under the combined transformation, justifying your answer. (2 marks)

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  6. Question 64 marks

    Equivalent single transformation

    WJEC Unit 1 (Non-calculator) — Higher

    Shape S is reflected in the x-axis to give shape S₁, then S₁ is reflected in the y-axis to give shape S₂.

    (a) The single transformation that maps S onto S₂ is a rotation. State the angle and centre of rotation. (3 marks)
    (b) Does this single rotation have any invariant points of S? Explain. (1 mark)

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    AI-generated · claude-opus-4-7 · v3-wjec-maths-leaves

Flashcards

G8 — Changes and invariance under rotation, reflection, translation, enlargement combinations

7-card SR deck for WJEC GCSE Mathematics (leaves batch 3) topic G8

7 cards · spaced repetition (SM-2)

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