2024 IGCSE Additional Mathematics May/June 0606/23 Q7

In the diagram, \(AD\) and \(BC\) are arcs of circles with common centre \(O\). \(ODC\) and \(OAB\) are straight lines with \(OA=5\,\mathrm{cm}\) and \(AB=4\,\mathrm{cm}\). Angle \(BOC=\theta\) radians. The area of the shaded region \(ABCD\) is \(4\pi\,\mathrm{cm}^2\).

1. Find \(\theta\).[3]
2. The straight line \(AC\) is added to the diagram and the region \(ACD\) is now shaded. Find the perimeter of the shaded region \(ACD\).[5]