In the diagram, $OAB$ is a sector of a circle with centre $O$ and radius $8 \mathrm{cm}$. Angle $BOA$ is $\alpha$ radians. $OAC$ is a semicircle with diameter $OA$.The area of the semicircle $OAC$ is twice the area of the sector $OAB$.Find $\alpha$ in terms of $\pi$.

In the diagram, shows triangle $ABC$ in which $AB$ is perpendicular to $BC$. The length of $AB$ is $4 \mathrm{cm}$ and angle $CAB$ is $\alpha$ radians. The arc $DE$ with centre $A$ and radius $2 \mathrm{cm}$ meets $AC$ at $D$ and $AB$ at $E$. Find, in terms of $\alpha$ the area of the shaded region.

In the diagram, shows triangle $ABC$ in which $AB$ is perpendicular to $BC$. The length of $AB$ is $4 \mathrm{cm}$ and angle $CAB$ is $\alpha$ radians. The arc $DE$ with centre $A$ and radius $2 \mathrm{cm}$ meets $AC$ at $D$ and $AB$ at $E$. Find, in terms of $\alpha$ the perimeter of the shaded region.

The diagram represents a metal plate $OABC$, consisting of a sector $OAB$ of a circle with centre $O$ and radius $r$, together with a triangle $OCB$ which is right-angled at $C$. Angle $AOB=\theta$ radians and $OC$ is perpendicular to $OA$. For the case where $r=10$ and $\theta =\frac{1}{5}\pi$, find the area of the plate.[Round off to one decimal places]

The diagram shows a sector $OAB$ of a circle with centre $O$ and radius $r$. Angle $AOB$ is $\theta$ radians.The point $C$ on $OA$ is such that $BC$ is perpendicular to $OA$.The point $D$ is on $BC$ and the circular arc $AD$ has centre $C$. Find $AC$ in terms of $r$ and $\theta$.

The diagram shows a circle with centre $A$ and radius $r$.Diameters $CAD$ and $BAE$ are perpendicular to each other. A larger circle has centre $B$ and passes through $C$ and $D$. Show that the radius of the larger circle is $r\sqrt{2}$.