AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 24

4 marks · Hard difficulty · Multi-step Problem

Given two similar triangles ABC and DEF, with side AC = 6 cm, angle C = 127°, side EF = 4.4 cm, angle F = 127°, and the area of ABC being 26.355 cm², calculate the area of triangle DEF.

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Question

Question 24 shows two similar triangles, ABC and DEF, marked as not drawn accurately. Triangle ABC has side AC labeled as 6 cm and the interior angle at vertex C labeled as 127 degrees. Triangle DEF is smaller, with angle F labeled as 127 degrees and side EF labeled as 4.4 cm. The text states that the area of ABC is 26.355 cm² and asks to work out the area of DEF for 4 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 24: First method mark M1 for using the area of a triangle formula 0.5 * 6 * x * sin(127) = 26.355 or rearranging to find BC = [10.98, 11.001]. Second method mark M1dep for calculating the linear scale factor, their [10.98, 11.001] ÷ 4.4 or 4.4 ÷ their [10.98, 11.001]. Third method mark M1dep for finding length DF and calculating the area of DEF using the sine rule, or applying the area scale factor (scale factor squared). Accuracy mark A1 for an answer in the range [4.2, 4.234]. Special case SC2 for [16.8, 16.94] if the 0.5 was omitted.

How to answer it

Area of Similar Triangles Using Trigonometry

📋 WHAT THIS QUESTION TESTS

This problem tests your ability to combine non-right-angled trigonometry with the properties of similar geometric shapes:

  • Using the sine formula for triangle area: Area = ½ab sin(C)
  • Rearranging an equation to find an unknown side length ( BC )
  • Identifying corresponding sides between similar shapes
  • Applying the linear scale factor ( k ) or the area scale factor ( k² )
QUESTION 24 (4 MARKS)

Finding the Area of Triangle DEF

AQA GCSE Higher Tier • Geometry & Trigonometry

💡 Key Formulae & Facts

  • Trigonometric Area:
    Area = ½ × a × b × sin(C)
    where C is the angle enclosed between sides a and b .
  • Similar Shapes:
    Triangles ABC and DEF have identical angles ( ∠C = ∠F = 127° ).
  • Scale Factor Relationships:
    If the linear scale factor between shapes is k , then the ratio of their areas is k² .

🧠 Exam Technique: Step-by-Step Approach

  • Method A (Area Scale Factor):
    1. Calculate length BC using the area formula.
    2. Compare corresponding sides BC and EF to find the linear scale factor k .
    3. Divide the area of ABC by k² .
  • Method B (Find Side DF):
    1. Find BC , calculate k , then find side DF = AC ÷ k .
    2. Compute Area = ½ × DF × EF × sin(127°) .

📐 Step-by-Step Solution (Method 1: Area Scale Factor)

Step 1: Calculate the missing side BC of triangle ABC

Area = ½ × AC × BC × sin(C)
26.355 = ½ × 6 × BC × sin(127°)
26.355 = 3 × sin(127°) × BC
BC = 26.355 ÷ (3 × sin(127°)) ≈ 11 cm (accepting [10.98, 11.001])

Step 2: Find the linear scale factor (k) between corresponding sides

Side BC corresponds to side EF ( 4.4 cm ).
Scale factor (k) = BC ÷ EF = 11 ÷ 4.4 = 2.5 (accepting [2.495, 2.50023])

Step 3: Calculate the area of triangle DEF using the area scale factor (k²)

Area scale factor = k² = 2.5² = 6.25
Area of DEF = Area of ABC ÷ k²
Area of DEF = 26.355 ÷ 6.25 = 4.2168 cm²

✅ Final Answer

4.2 cm² (or any value in the range 4.2 to 4.234)

Mark Scheme Breakdown:
• M1: Correct setup to find side BC: 0.5 × 6 × BC × sin(127) = 26.355
• M1 (dep): Finding length scale factor BC ÷ 4.4 (approx 2.5)
• M1 (dep): Applying area scale factor: 26.355 ÷ (2.5)² OR finding side DF = 2.4 cm and evaluating ½ × 4.4 × 2.4 × sin(127)
• A1: Final answer in range [4.2, 4.234]

❌ Common Misconceptions & Traps

  • Forgetting the ½ in the sine rule: Forgetting ½ in ½ab sin(C) leads to BC = 5.5 and an answer of 16.8 - 16.94 . This gives a special concession of at most SC2.
  • Dividing area by the linear scale factor: Dividing 26.355 ÷ 2.5 = 10.54 instead of squaring the scale factor ( 2.5² = 6.25 ). Remember: lengths scale by k, areas scale by k²!
  • Premature rounding: Rounding BC or sin(127°) too early causes rounding inaccuracies in the final area. Keep values stored in your calculator memory.
  • Calculator in radians: Ensure your calculator is in DEG mode; otherwise sin(127) gives an incorrect negative value.

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 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.