Answer: (b) \(2.25\) radians; (c) \(85.9\text{ cm}^2\).

Answer: (b) \(2.25\) radians; (c) \(85.9\text{ cm}^2\).

(a) In triangle \(AOB\), \(OA=OB=8\) and chord \(AB=10\). Bisecting the chord gives a right triangle with opposite side \(5\) and hypotenuse \(8\).

So

\(\sin\left(\frac{\angle BOA}{2}\right)=\frac58.\)

Therefore

\(\angle BOA=2\sin^{-1}\left(\frac58\right)=1.350263\ldots.\)

Hence \(\angle BOA=1.35\) radians correct to \(2\) decimal places.

(b) Arc length is given by \(s=r\theta\). Since the minor arc \(BC\) has length \(18\),

\(18=8\theta.\)

Thus

\(\angle BOC=\theta=\frac{18}{8}=2.25.\)

(c) The minor angle \(AOC\) is the remaining angle around the circle:

\(\angle AOC=2\pi-1.350263\ldots-2.25=2.68292\ldots.\)

The sector area is

\(\frac12r^2\theta=\frac12(8^2)(2.68292\ldots).\)

So the area is

\(85.9\text{ cm}^2\)

to \(3\) significant figures.