AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 23

Two-Step Pythagoras' Theorem with Shared Sides

📋 What this question tests

Identifying right-angled triangles: Spotting the two separate right-angled triangles sharing a common vertical boundary.
Finding a shorter side: Rearranging Pythagoras' theorem (a² = c² − b²) when given the hypotenuse.
Finding the hypotenuse: Applying Pythagoras' theorem (c² = a² + b²) to find the unknown length x.
Mathematical reasoning ("Show that"): Demonstrating clearly that the final result lies strictly between 10 and 11, either by evaluating the square root to a decimal or by comparing square numbers.

Question Walkthrough (4 Marks)

Use Pythagoras' theorem to show that the value of x is between 10 and 11

💡 Key Knowledge

Pythagoras' Theorem: For any right-angled triangle, a² + b² = c² , where c is the hypotenuse (the longest side opposite the 90° angle).
Finding a shorter side: shorter side = √(hypotenuse² − other side²) .
Finding the hypotenuse: hypotenuse = √(side₁² + side₂²) .
Pythagorean Triple: Recognising 5, 12, 13 saves time and provides an instant check for the middle length.

🧠 Exam Technique

Label the common side: Give the vertical line a label such as h or y, or write the calculated value directly onto the diagram.
Work sequentially: You cannot find x directly. Always start with the triangle that has two known lengths (the right-hand triangle).
Write clear concluding statements: For a "show that" question, complete the argument by either stating 10.3 is between 10 and 11 or showing 100 < 106 < 121 .
Avoid premature rounding: Keep exact values (or surds) until the final evaluation step.

📐 Step-by-Step Calculation

Step 1: Calculate the common vertical side (right-hand triangle)

The right-hand triangle has a right angle at the top, hypotenuse = 13 cm , and horizontal side = 12 cm .
Let the vertical height be h:
h² = 13² − 12²
h² = 169 − 144 = 25
h = √25 = 5 cm

Awarded M1 for 12² or 13² (144 or 169), and M1 for finding 5 (or √25).

Step 2: Use the vertical side to find x (left-hand triangle)

The left-hand triangle has sides adjacent to the right angle measuring 9 cm and 5 cm .
The side x is the hypotenuse:
x² = 9² + 5²
x² = 81 + 25 = 106
x = √106 ≈ 10.2956... cm (or 10.3 cm to 1 d.p.)

Awarded M1 for 9² + (their 5)² or 81 + 25 .

Step 3: Complete the "show that" conclusion

You can conclude using either of two fully credited methods:

Method A (Decimal value): x = 10.3 (or any value in the range [10.2, 10.3]), and state that 10.3 is between 10 and 11.
Method B (Comparing squares): Since 10² = 100 and 11² = 121 , and 106 lies between 100 and 121, √106 must be between 10 and 11.

Awarded A1 for complete, correct mathematical working showing x is between 10 and 11.

✅ Model Answer

Vertical side² = 13² − 12² = 169 − 144 = 25
Vertical side = √25 = 5 cm

x² = 9² + 5²
x² = 81 + 25 = 106
x = √106 = 10.3 cm (to 1 d.p.)

Since 10 < 10.3 < 11, the value of x is between 10 and 11.

❌ Common Errors & Examiner Traps

Adding instead of subtracting: Computing 13² + 12² = 313 because students forget that 13 cm is already the hypotenuse.
Misidentifying the hypotenuse: Subtracting in the left-hand triangle (e.g. 9² − 5² ) instead of adding to find x.
Poor notation / missing steps: Writing 9² + 5² = √106 = 10.3 . This loses marks because 9² + 5² is 106, not √106! Keep lines separate: x² = 106 then x = √106 .
Using trigonometry or measuring: The mark scheme explicitly states M0 for scale drawing or trigonometric methods alone when Pythagoras is specified.

AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 23

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