Write down the exact value of $\theta$ in radians.

Find the value of $r$.

Find AB.

$\theta  = \frac{{2\pi }}{5}$     A1     N1

[1 mark]

correct expression for area     (A1)

eg$\,\,\,\,\,$$A = \frac{1}{2}{r^2}\left( {\frac{{2\pi }}{5}} \right),{\text{ }}\frac{{\pi {r^2}}}{5}$

evidence of equating their expression to $20\pi$     (M1)

eg$\,\,\,\,\,$$\frac{1}{2}{r^2}\left( {\frac{{2\pi }}{5}} \right) = 20\pi ,{\text{ }}{r^2} = 100,{\text{ }}r =  \pm 10$

$r = 10$    A1     N2

[3 marks]

METHOD 1

evidence of choosing cosine rule     (M1)

eg$\,\,\,\,\,$${a^2} = {b^2} + {c^2} - 2bc\cos A$

correct substitution of their $r$ and $\theta$ into RHS     (A1)

eg$\,\,\,\,\,$${10^2} + {10^2} - 2 \times 10 \times 1{\text{0}}\cos \left( {\frac{{2\pi }}{5}} \right)$

11.7557

${\text{AB}} = 11.8{\text{ (mm)}}$     A1     N2

METHOD 2

evidence of choosing sine rule     (M1)

eg$\,\,\,\,\,$$\frac{{\sin A}}{a} = \frac{{\sin B}}{b}$

correct substitution of their $r$ and $\theta$     (A1)

eg$\,\,\,\,\,$$\frac{{\sin \frac{{2\pi }}{5}}}{{{\text{AB}}}} = \frac{{\sin \left( {\frac{1}{2}\left( {\pi  - \frac{{2\pi }}{5}} \right)} \right)}}{{10}}$

11.7557

${\text{AB}} = 11.8{\text{ (mm)}}$     A1     N2

[3 marks]