AQA GCSE Mathematics Paper 2 (Foundation), June 2025: Question 10

3 marks · Easy difficulty · Short Answer

Given two mathematically similar shapes, determine an unknown corresponding angle and find an unknown side length using scale factors.

Practise this question

Question

Two similar polygons are displayed labeled 'Not drawn accurately'. The larger polygon has a horizontal base of 15 cm, a vertical left edge labeled x, and an internal acute angle labeled 30 degrees. The smaller polygon has a horizontal base of 6 cm, a vertical left edge of 3 cm, and a corresponding acute angle labeled y. Part (a) asks to write down the value of y for 1 mark, and part (b) asks to work out the value of x for 2 marks.

Mark scheme

Show the mark scheme Mark scheme for question 10. Part (a) awards B1 for 30, with guidance noting further work like 1/4 * 30 = 7.5 scores B0. Part (b) awards M1 for finding a scale factor or ratio such as 15 / 6, 2.5, 6 / 3, 2, 6 / 15, 0.4, 3 / 6, 0.5, or equation x / 15 = 3 / 6. A1 is awarded for 7.5, 7 1/2, or 15/2.

How to answer it

Similar Shapes: Angles and Side Lengths

📌 What this question tests

This question assesses your understanding of mathematical similarity in 2D shapes:

  • Angle preservation: Corresponding angles in mathematically similar shapes are always strictly equal.
  • Scale factors: Corresponding side lengths are proportional and related by a single linear multiplier (scale factor).
Question 10 (a)

Finding the Missing Angle y

1 Mark • Target: Quick recall and recognition

✅ Correct Answer

y = 30°

B1: Independent mark for exactly 30 (degree symbol is already on the answer line).

💡 Key Knowledge

  • When two shapes are similar, their side lengths change, but their angles never change.
  • Angle y corresponds directly to the given angle of 30° in the larger shape.
  • No calculation is needed: simply write down the matching angle!

❌ Common Errors & Misconceptions

  • Scaling the angle: Trying to divide or multiply the angle by a scale factor (e.g. calculating 30 ÷ 2.5 = 12° or ¼ × 30 = 7.5° ). The mark scheme specifically awards B0 if any extra calculations alter the angle.
  • Thinking smaller shape = smaller angles: Angles determine the shape; enlarging or reducing a shape leaves all angles completely identical.

🧠 Exam Technique

Notice the command words: "Write down" means no working is required. Whenever you see "Write down" for an angle on a similar shapes question, look directly for the matching angle on the other shape.

Question 10 (b)

Calculating the Missing Side Length x

2 Marks • Target: Method and accuracy with scale factors

📐 Step-by-Step Calculation

Method 1: Using the scale factor between shapes

  1. Find the linear scale factor:
    Divide corresponding known sides:
    Scale Factor = 15 ÷ 6 = 2.5
  2. Multiply to find the unknown length:
    The vertical height on the smaller shape is 3 cm :
    x = 3 × 2.5 = 7.5 cm

Method 2: Using internal shape ratios

  1. On the small shape, compare base to height:
    6 ÷ 3 = 2 (the base is twice the height).
  2. Apply the same ratio to the large shape:
    x = 15 ÷ 2 = 7.5 cm

✅ Acceptable Final Answers

7.5 or 7 ½ or 15/2

M1 (Method): For any valid scale factor or ratio statement:
  • 15 ÷ 6 or 2.5
  • 6 ÷ 3 or 2
  • 3 ÷ 6 or 0.5
  • x / 15 = 3 / 6
A1 (Accuracy): 7.5 , 7 ½ , or 15/2

❌ Common Errors

  • Additive reasoning: Calculating the difference 15 - 6 = 9 and adding 3 + 9 = 12 . Similarity is multiplicative, not additive!
  • Inverted division: Doing 3 ÷ 2.5 = 1.2 cm instead of multiplying. Always do a sanity check: shape x is clearly the larger shape, so x must be greater than 3 cm .
  • Pairing the wrong sides: Dividing 15 ÷ 3 . Make sure you match horizontal base with horizontal base ( 15 and 6 ), and vertical side with vertical side ( x and 3 ).

🧠 Exam Technique

  • Always show the division: Simply writing 15 ÷ 6 guarantees the M1 method mark, even if you make a slip in your mental arithmetic later.
  • Diagram credit: If you write 7.5 directly on the diagram next to x , examiners will award marks as long as it isn't contradicted by conflicting working in the answer space.

Topics

Geometry and measures · Ratio, proportion and rates of change · 3.3 Ratio, proportion and rates of change · 3.4.2 Mensuration and calculation

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