Answer: \(\angle AOC=1.287\) radians; perimeter \(32.4\text{ cm}\); area \(28.1\text{ cm}^2\).

(a) In triangle \(AOC\), \(OA=OC=5\) and \(AC=6\).

Bisect the chord \(AC\). Then

Hence

So \(\angle AOC=1.287\) radians to 3 decimal places.

(b) Since \(AOB\) and \(COD\) are diameters, the angle \(DOB\) is equal to angle \(AOC\). Therefore the arc \(DB\) has length

\(5(1.2870\ldots).\)

The shaded perimeter consists of the triangle perimeter \(AO+OC+AC\), together with the sector perimeter \(OD+OB+\text{arc }DB\). Therefore

\(P=(5+5+6)+(5+5+5(1.2870\ldots)).\)

This gives

\(P=32.435\ldots\text{ cm}\),

so the perimeter is \(32.4\text{ cm}\) to 3 significant figures.

(c) The shaded area is the area of triangle \(AOC\) plus the area of sector \(DOB\):

This is

\(28.1\text{ cm}^2\)

to 3 significant figures.