The diagram shows triangle ABC - OCR - GCSE Maths - Question 17 - 2021 - Paper 1

To find the length of AB, we can use the Law of Cosines. The Law of Cosines states: $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

In our case, we have:

• Side AC (b) = 48 mm
• Side BC (a) = 85 mm
• Angle BAC (C) = 53°

Let the length of AB be c. Based on the Law of Cosines, the formula becomes: $AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(53°)$

Substituting the known values: $AB^2 = 48^2 + 85^2 - 2 \cdot 48 \cdot 85 \cdot \cos(53°$

Calculating the squares: $= 2304 + 7225 - 8160 \cdot \cos(53°$

Calculating (\cos(53°)) gives approximately 0.6018:

So, substituting: $AB^2 = 2304 + 7225 - 8160 \cdot 0.6018$ $= 2304 + 7225 - 4907.56$ $= 4615.44$

Now take the square root to find AB: $AB = \sqrt{4615.44} \approx 67.96$ mm.

Hence, the length of AB is approximately 68 mm.