Triangle and sector combined - algebraic/general expressions

8 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-10_348_700_262_721} In the diagram, \(A B C\) is an isosceles triangle with \(A B = B C = r \mathrm {~cm}\) and angle \(B A C = \theta\) radians. The point \(D\) lies on \(A C\) and \(A B D\) is a sector of a circle with centre \(A\).

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 10\) and \(\theta = 0.6\), find the perimeter of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-3_625_897_260_623} The diagram shows triangle \(A B C\) in which \(A B\) is perpendicular to \(B C\). The length of \(A B\) is 4 cm and angle \(C A B\) is \(\alpha\) radians. The arc \(D E\) with centre \(A\) and radius 2 cm meets \(A C\) at \(D\) and \(A B\) at \(E\). Find, in terms of \(\alpha\),

1. the area of the shaded region,
2. the perimeter of the shaded region.

\includegraphics{figure_2} In the diagram, \(AYB\) is a semicircle with \(AB\) as diameter and \(OAXB\) is a sector of a circle with centre \(O\) and radius \(r\). Angle \(AOB = 2\theta\) radians. Find an expression, in terms of \(r\) and \(\theta\), for the area of the shaded region. [4]

A sector of a circle has radius \(r\) cm and sector angle \(\theta\) radians. It is divided into two regions, A and B. Region A is an isosceles triangle with the equal sides being of length \(a\) cm, as shown in Fig. 6. \includegraphics{figure_6}

1. Express the area of B in terms of \(a\), \(r\) and \(\theta\). [2]
2. Given that \(r = 12\) and \(\theta = 0.8\), find the value of \(a\) for which the areas of A and B are equal. Give your answer correct to 3 significant figures. [2]