IB AA HL May 2026 P2 TZC Q4

Two circles have a common centre, \(O\). Points \(A\) and \(B\) lie on the circumference of the larger circle, which has radius \(OA=R\). Points \(C\) and \(D\) lie on \(OB\) and \(OA\) respectively, and they also lie on the circumference of the smaller circle. The smaller circle has radius \(OC=r\) and the acute angle \(D\hat{O}C=A\hat{O}B=\theta\) radians.

The area of the sector \(OAB\) is \(50\text{ cm}^2\) and its perimeter is \(40\text{ cm}\). The area of sector \(ODC\) is \(18\text{ cm}^2\).

1. Find the value of \(\theta\).[5]
2. Hence, find the value of \(r\).[2]