AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 18

4 marks · Medium difficulty · Multi-step Problem

Calculate the length of line segment AD formed by two intersecting line segments using angle properties of isosceles triangles and the cosine rule.

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Question

Geometric diagram showing two intersecting lines forming two triangles, triangle ADX and triangle BCX, intersecting at a central vertex. In the lower triangle, the two side segments from the intersection are both labelled 4.1 cm, and the base angle at vertex C is labelled 32°. In the upper triangle, the sides extending to A and D are labelled 9.6 cm and 2.8 cm, respectively. The question asks to work out the length of line segment AD.

Mark scheme

Show the mark scheme Mark scheme for question 18 showing 4 total marks: B1 for calculating the angle 116 degrees at the central intersection (seen in CXB or AXD); M1 for setting up the cosine rule formula 9.6 squared plus 2.8 squared minus 2 times 9.6 times 2.8 times cos(116), giving [123.5, 123.7]; M1dep for taking the square root of that value; A1 for an answer in the range [11.1, 11.1221] or 11 with B1M2 awarded.

How to answer it

Question 18: Length of AD via Angle Rules and Cosine Rule

📌 What This Question Tests

This question assesses multi-step geometric problem solving on the Higher Tier paper, specifically combining:

  • Identifying and using the properties of an isosceles triangle (base angles are equal).
  • Applying angle sum in a triangle (angles add up to 180°) and vertically opposite angles.
  • Applying the Cosine Rule for finding a side: a² = b² + c² - 2bc cos(A) in a non-right-angled triangle.
  • Correct calculator execution with obtuse angles and negative trigonometric values.

Question Walkthrough & Solution

Calculate the side length AD [4 Marks]

💡 Key Geometric Knowledge

  • Label the intersection: Let the point where line segments AB and CD cross be called point X.
  • Isosceles triangle rule: Triangle CXB has two sides of length 4.1 cm. Thus, base angles are equal:
    Angle XBC = Angle XCB = 32° .
  • Vertically opposite angles: Angle AXD = Angle CXB .
  • Cosine Rule (finding a side):
    a² = b² + c² - 2bc cos(A)
    Used when you know two sides and the included angle (SAS).

🧠 Exam Technique & Strategy

  • Bridge the two triangles: Information is given in the bottom triangle ( CXB ) and required in the top triangle ( AXD ). The bridge is the angle at the shared vertex X .
  • Beware of obtuse cosines: For Angle AXD = 116° , cos(116°) is negative (-0.4384). Subtracting a negative becomes adding!
  • Keep full precision: Do not round intermediate values on your calculator until writing the final answer.

📐 Step-by-Step Calculation

  1. Find the missing angle in the bottom triangle (CXB):
    Triangle CXB is isosceles because side CX = 4.1 cm and side BX = 4.1 cm .
    Therefore, angle CBX = 32° .
    Angle CXB = 180° - 32° - 32° = 116° .
  2. Find the included angle in the top triangle (AXD):
    Angles CXB and AXD are vertically opposite.
    Therefore, angle AXD = 116° .
  3. Set up the Cosine Rule in triangle AXD:
    Given: AX = 9.6 cm , DX = 2.8 cm , included angle AXD = 116° .
    AD² = 9.6² + 2.8² - 2(9.6)(2.8)cos(116°)
  4. Calculate the value of AD²:
    9.6² = 92.16
    2.8² = 7.84
    2 × 9.6 × 2.8 = 53.76
    AD² = 92.16 + 7.84 - 53.76 × cos(116°)
    AD² = 100 - 53.76 × (-0.43837...)
    AD² = 100 + 23.5668... = 123.567...
  5. Take the square root to find AD:
    AD = √(123.567...) = 11.116... cm

✅ Correct Final Answer

11.1 cm (or any value in the range [11.1, 11.1221])

Mark Breakdown:
  • B1: Finding 116° (may be shown at angle CXB or AXD ).
  • M1: Correct substitution into the cosine rule: 9.6² + 2.8² - 2(9.6)(2.8)cos(their 116) or evaluating to range [123.5, 123.7] .
  • M1 (dep): Taking the square root of their value: √(their [123.5, 123.7]) .
  • A1: Final answer in the range [11.1, 11.1221] (accept 11 if B1 M2 previously awarded).

❌ Common Mistakes & Traps

  • Missing the isosceles triangle: Failing to spot that both segments are 4.1 cm , leading to an incorrect or assumed angle.
  • Sign error with cos(116°): Because cos(116°) < 0 , students often incorrectly calculate 100 - 23.57 = 76.43 instead of 100 - (-23.57) = 123.57 .
  • Order of operations error (BIDMAS): Combining (92.16 + 7.84 - 53.76) × cos(116°) , which computes 46.24 × cos(116°) . Multiplication must be done before subtraction!
  • Forgetting the square root: Leaving the answer as 123.6 (which is AD² , not AD ). A quick check against the diagram shows a line of length 123.6 cm would be absurdly long!

Topics

Geometry and measures · 3.4.1 Properties and constructions · 3.4.2 Mensuration and calculation

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.