In the diagram, a lighthouse is at \(O\). Four ships, \(A\), \(B\), \(C\) and \(D\), are positioned around it with \(OA=3\text{ km}\), \(OB=5\text{ km}\), \(OC=8\text{ km}\) and \(OD=6\text{ km}\). The bearing of \(A\) from \(O\) is \(027^\circ\) and the bearing of \(B\) from \(O\) is \(103^\circ\). Ship \(D\) lies due west of \(O\) and Ship \(C\) lies due south of \(D\).
Calculate the distance between Ship \(A\) and Ship \(B\).[3]
The keeper of the lighthouse is standing on top of the lighthouse, which is 20 m tall. Calculate the angle of depression, from where he is standing to Ship \(A\).[2]
Calculate the bearing of Ship \(C\) from \(O\).[2]
A speedboat travels directly from Ship \(A\) to Ship \(B\). Calculate the shortest distance between the speedboat and the lighthouse during its journey.[2]
It is suggested that Ships \(A\), \(B\) and the lighthouse at \(O\) form a right-angled triangle. Showing your working clearly, determine whether this is true.[2]