AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 4

4 marks · Medium difficulty · Multi-step Problem

Work out the perimeter of an equilateral triangle whose sides are given algebraically as 6x - 8 and 2x + 12.

Practise this question

Question

An equilateral triangle with two side lengths indicated algebraically: one side is labelled 6x minus 8 and another side is labelled 2x plus 12. Text states all measurements are in centimetres and asks to work out the value of the perimeter of the triangle.

Mark scheme

Show the mark scheme Mark scheme for Question 4 awarding 4 marks: M1 for setting up the equation 6x - 8 = 2x + 12; M1dep for rearranging to 4x = 20 or finding x = 5; M1dep for calculating a side length by substituting x = 5 into either expression to get 22; A1 for the final answer 66.

How to answer it

Perimeter of an Equilateral Triangle with Unknowns

What this question tests
  • Geometric properties: Recognizing that an equilateral triangle has 3 equal sides and 3 equal angles.
  • Forming equations: Equating two algebraic expressions for the side lengths ( 6x − 8 = 2x + 12 ).
  • Solving linear equations: Rearranging and isolating terms with unknown variables on both sides.
  • Multistep substitution: Finding the side length and calculating total perimeter ( side × 3 ).

Question 4 Breakdown (4 Marks)

Forming and solving an algebraic equation to find perimeter

💡 Key Knowledge

  • An equilateral triangle has 3 sides of exactly equal length.
  • Since both expressions describe sides of the same equilateral triangle:
    Side 1 = Side 2
  • Perimeter is the total distance around the outside edge:
    Perimeter = Side + Side + Side = 3 × Side

🧠 Exam Technique

  • Always state the initial equation clearly: Writing 6x − 8 = 2x + 12 secures the first method mark instantly.
  • Self-check by substitution: Test your value of x in both expressions to verify that both sides equal the same number.
  • Reread the final demand: The question asks for the perimeter, not just x and not just one side length!

📐 Step-by-Step Solution

Step 1: Form the equation using equilateral properties

Since all sides are equal in length:

6x − 8 = 2x + 12

awarded M1 for setting up the equation in terms of x.

Step 2: Collect terms to solve for x

Subtract 2x from both sides:

4x − 8 = 12

Add 8 to both sides:

4x = 20

Divide by 4 :

x = 5

awarded M1 (dep) for collecting terms correctly or finding x = 5 .

Step 3: Work out the length of one side

Substitute x = 5 into either expression:

Side = 6(5) − 8 = 30 − 8 = 22 cm

Check with the other expression: 2(5) + 12 = 10 + 12 = 22 cm (Correct!)

awarded M1 (dep) for substituting their value of x to find side length (22).

Step 4: Calculate the perimeter

An equilateral triangle has 3 equal sides:

Perimeter = 22 + 22 + 22 = 3 × 22 = 66 cm

awarded A1 for the correct final answer of 66.

❌ Common Errors to Avoid

  • Equating to 180°: Writing (6x − 8) + (2x + 12) = 180 . These expressions represent lengths in cm, not interior angles! This scores 0 marks (M0).
  • Stopping at x = 5: Forgetting what the question asks. Solving for x is only halfway through the task.
  • Stopping at side = 22: Calculating the length of a side but forgetting to multiply by 3 to find the perimeter.
  • Adding only the two given sides: Adding 22 + 22 = 44 , forgetting that a triangle has three sides.

✅ Final Mark Scheme Summary

  • M1: 6x − 8 = 2x + 12
  • M1dep: 4x = 20 or x = 5
  • M1dep: 6 × 5 − 8 or 2 × 5 + 12 or 22
  • A1: 66 (cm)

Topics

Algebra · Geometry and measures · 3.2.3 Solving equations and inequalities · 3.2.1 Notation, vocabulary and manipulation · 3.4.1 Properties and constructions

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.