Given \(\Delta \)ABC, with lengths shown in the diagram below, find the length of the line segment [CD].

METHOD 1

\(\frac{{\sin C}}{7} = \frac{{\sin 40}}{5}\)     M1(A1)

\({\text{B}}\hat {\text{C}}{\text{D}} = 64.14...^\circ \)     A1

\({\text{CD}} = 2 \times 5\cos 64.14...\)     M1

Note: Also allow use of sine or cosine rule.

\({\text{CD}} = 4.36\)     A1

[5 marks]

 METHOD 2

let \({\text{AC}} = x\)

cosine rule

\({5^2} = {7^2} + {x^2} - 2 \times 7 \times x\cos 40\)     M1A1

\({x^2} - 10.7 ... x + 24 = 0\)

\(x = \frac{{10.7... \pm \sqrt {{{\left( {10.7...} \right)}^2} - 4 \times 24} }}{2}\)     (M1)

\(x = 7.54\); \(3.18\)     (A1)

CD is the difference in these two values \(= 4.36\)     A1

Note: Other methods may be seen.

[5 marks]

This was an accessible question to most candidates although care was required when calculating the angles. Candidates who did not annotate the diagram or did not take care with the notation for the angles and sides often had difficulty recognizing when an angle was acute or obtuse. This prevented the candidate from obtaining a correct solution. There were many examples of candidates rounding answers prematurely and thus arriving at a final answer that was to the correct degree of accuracy but incorrect.