Past Paper Example 1
AQA Jun 2025 3F Q23
4 marks
Double right-angled triangles sharing common side. Right triangle has hypotenuse 13 cm, base 12 cm. Left triangle has base 9 cm, hypotenuse x cm. Show 10 < x < 11. [4 marks]
Use Pythagoras’ theorem to show that the value of x is between 10 and 11
Worked Solution & Mark Scheme (4 marks)
Right triangle has hypotenuse 13 cm and side 12 cm: M1
Shared vertical side h = √(13² – 12²) = √(169 – 144) = √25 = 5 cm M1
Left triangle has sides 9 cm and 5 cm with hypotenuse x:
x² = 9² + 5² = 81 + 25 = 106 ⇒ x = √106 M1
Since 10² = 100 and 11² = 121, and 100 < 106 < 121, we have 10 < √106 < 11
Thus x is between 10 and 11. C1
Shared vertical side h = √(13² – 12²) = √(169 – 144) = √25 = 5 cm M1
Left triangle has sides 9 cm and 5 cm with hypotenuse x:
x² = 9² + 5² = 81 + 25 = 106 ⇒ x = √106 M1
Since 10² = 100 and 11² = 121, and 100 < 106 < 121, we have 10 < √106 < 11
Thus x is between 10 and 11. C1
Answer: Shared side = 5 cm, x = √106 ≈ 10.3 cm, which is between 10 (√100) and 11 (√121)