AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 22

3 marks · Medium difficulty · Short Answer

Describe the single transformation that maps shape A to shape B on the coordinate grid.

Practise this question

Question

A Cartesian grid with x and y axes from 0 to 10. Shape A is a right-angled triangle with vertices at (6, 1), (10, 1), and (10, 7). Shape B is an inverted right-angled triangle with vertices at (1, 4), (1, 7), and (3, 7). The question asks to describe the single transformation that maps shape A to shape B.

Mark scheme

Show the mark scheme Mark scheme table for question 22: Award B1 for 'Enlarge(ment)'. Award B1 for scale factor '-1/2' (condone minus half). Award B1 for centre of enlargement '(4, 5)' (condone missing brackets; do not accept vector notation (4 over 5)). Notes state combined transformation scores B0B0B0, but 'Rotation, -1/2, about (4, 5)' earns B0B1B1.

How to answer it

Describing a Negative Fractional Enlargement

📋 WHAT THIS QUESTION TESTS

This question tests your ability to fully describe a single transformation mapping one shape onto another on a coordinate grid. Key skills include recognising that a change in size combined with an inverted orientation represents a negative enlargement, calculating a fractional scale factor, and determining the exact coordinates of the centre of enlargement using ray lines.

Question 22 (3 Marks)

Describe the single transformation that maps shape A to shape B

✅ Full Mark Answer

Enlargement, scale factor -½ (or -0.5), centre of enlargement (4, 5).

Mark Scheme Breakdown:
• [B1] State "Enlargement" (or "Enlarge")
• [B1] State scale factor -½ or -0.5
• [B1] State centre (4, 5) (coordinate format)

💡 Key Knowledge

  • Enlargement with negative scale factor: Inverts the shape (turns it upside down and back-to-front) and projects it across to the opposite side of the centre of enlargement.
  • Fractional scale factor: Because the shape gets smaller ( |scale factor| < 1 ), the scale factor is a fraction or decimal between 0 and 1 (or 0 and -1).
  • Direction matters: The question asks for the transformation from A to B, meaning A is the original object and B is the image.

📐 Step-by-Step Method

  1. Identify the transformation type: Shape B is smaller than shape A and has been inverted. Because the size changes, it must be an Enlargement.
  2. Calculate the scale factor: Compare corresponding side lengths:
    • Horizontal base of A = 10 - 6 = 4 units
    • Horizontal top of B = 3 - 1 = 2 units
    • Scale ratio = Image length / Object length = 2 / 4 = ½
    Since the shape is inverted, the scale factor is negative: -½.
  3. Find the centre of enlargement: Join corresponding vertices with straight lines (ray lines) using a ruler:
    • Right-angle vertex on A (10, 1) maps to right-angle vertex on B (1, 7) .
    • Top vertex on A (10, 7) maps to bottom vertex on B (1, 4) .
    • Sharp left vertex on A (6, 1) maps to right vertex on B (3, 7) .
    Where all lines intersect is the centre: (4, 5).

🧠 Exam Technique & Examiner Insight

  • Use a sharp pencil and ruler: Draw at least two ray lines connecting matching vertices all the way through the grid. The point where they cross gives you the coordinates of the centre directly.
  • Check midpoint behaviour: For a scale factor of -1 , the centre is the exact midpoint. For -½ , the centre is twice as close to B as it is to A.
    From (4, 5) to (1, 7) is (-3, +2) .
    From (4, 5) to (10, 1) is (+6, -4) .
    Notice that (-3, +2) = -½ × (6, -4) . This confirms (4, 5) is correct!

❌ Common Errors & Pitfalls

  • Giving two transformations (e.g. "Rotation then reduction"): The question explicitly says single transformation. If you write two transformations (such as "Rotation and Enlargement"), you immediately score 0 marks.
  • Forgetting the negative sign: Stating ½ instead of -½ loses the scale factor mark. The inverted orientation requires a negative sign.
  • Inverting the scale factor: Writing -2 because shape A has side lengths twice as big. Remember it is always Image ÷ Object (B ÷ A), which is 2 ÷ 4 = ½ .
  • Writing the centre as a column vector: A centre of enlargement is a position on the grid, so write it as coordinate brackets (4, 5) , NOT a vector (⁴₅) . The mark scheme specifically rejects column vectors for this mark!

Topics

Geometry and measures · 3.4.1 Properties and constructions

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.