AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 16

Perimeter of an Equilateral Triangle Using Algebra

📋 What this question tests

This 4-mark question tests your ability to connect geometric properties with algebraic problem-solving. Specifically, you need to:

Recall that all three sides of an equilateral triangle are equal in length.
Set up and solve a linear equation with unknowns on both sides ( ax + b = cx + d ).
Substitute the value of the variable back into an expression to find a side length.
Calculate the total perimeter by adding all three equal side lengths (or multiplying one side by 3).

Question 16 Walkthrough

Total Marks: 4 | Calculator / Non-Calculator

💡 Key Knowledge

Equilateral Triangle: All 3 sides are identical in length, and all 3 interior angles are 60°.
Perimeter: The total distance around the outside edge ( Perimeter = Side 1 + Side 2 + Side 3 ).
Forming Equations: Because the side lengths are equal, we can write:
6x − 8 = 2x + 12

🧠 Exam Technique & Strategy

Always define the geometric rule first: Don't just stare at the expressions—ask yourself what the word "equilateral" tells you about the sides.
Check both sides match: After finding x , substitute it into both given expressions. If they give different numbers, you made an error in your algebra!
Answer the actual question: Finding x = 5 or side length 22 is not the end; you must calculate the perimeter.

📐 Step-by-Step Calculation

1 Form an equation equating two sides:

Since the triangle is equilateral, both given sides are equal:
6x − 8 = 2x + 12

2 Solve the linear equation for x:

Subtract 2x from both sides:
4x − 8 = 12
Add 8 to both sides:
4x = 20
Divide by 4 :
x = 5

3 Find the length of one side:

Substitute x = 5 into either side expression:
Using Side 1: 6(5) − 8 = 30 − 8 = 22 cm
Self-Check using Side 2: 2(5) + 12 = 10 + 12 = 22 cm ✓ (Both sides match!)

4 Calculate the total perimeter:

An equilateral triangle has 3 sides of equal length:
Perimeter = 22 + 22 + 22 = 3 × 22 = 66 cm

✅ Correct Answer & Mark Scheme Breakdown

Perimeter = 66 cm

Mark Allocation:

M1: Setting up a correct algebraic equation equating both side lengths: 6x − 8 = 2x + 12 (or equivalent in another letter).
M1 (dep): Correctly rearranging to collect terms and solve for x: 4x = 20 or x = 5 .
M1 (dep): Substituting their value of x into an expression to find a side length ( 22 ), or writing an expression for all sides (e.g. 14x − 4 with x = 5 substituted).
A1: Correct final perimeter of 66.

❌ Common Misconceptions & Traps

Confusing side lengths with angle sums: Some candidates set (6x − 8) + (2x + 12) = 180 . The given algebraic expressions represent lengths in cm, NOT angle measurements! The mark scheme awards 0 marks (M0) for this approach.
Stopping too early at x = 5: Solving for x is only halfway through the problem. Always re-read the final line: "Work out the value of the perimeter..."
Stopping at one side length (22 cm): Finding 22 cm is a crucial milestone, but you must remember to multiply by 3 to obtain the complete boundary perimeter.
Negative sign errors when rearranging: When moving −8 to the right-hand side, remember to add 8: 12 + 8 = 20 , not 12 − 8 = 4 .

AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 16

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