2026 CSS P1 Q10

The diagram shows a pentagon \(ABCEF\) formed by a rectangle \(ABDF\) and two isosceles triangles \(BCD\) and \(DEF\) where \(CDE\) is a straight line. The angle \(EDF\) is \(\theta\), \(BC=CD=3\) cm and \(DE=EF=4\) cm.

1. Show that \(P\) cm, the perimeter of the pentagon, can be expressed in the form \(14+6\sin\theta+8\cos\theta\).[2]
2. Express \(P\) in the form of \(k+R\sin(\theta+\alpha)\), where \(R>0\) and \(\alpha\) is an acute angle.[2]
3. Write down the maximum possible value of \(P\) and find the value of \(\theta\) at which this maximum value occurs.[2]
4. Find the value(s) of \(\theta\) for which \(P=23\) cm.[3]