Geometric Shapes and Quadratic Equations

In the diagram, the circle with centre at \(B\) has a radius \(\left( 2r+1 \right)\)cm.

The semicircle with centre at \(A\) and the semicircle with centre at \(C\) are identical and each has a radius \(\left( 12-6r \right)\)cm.

\(O\) is the centre of the largest semicircle.

1. Write down an expression, in terms of \(r\), for
   1. \(BC\),
   2. \(BO\).
2. By using Pythagoras Theorem, form an equation in \(r\) and show that it reduces to \(18{{r}^{2}}-57r+42=0\).
3. Solve the equation \(18{{r}^{2}}-57r+42=0\) to find the possible value of \(r\).
4. Find the area of the unshaded region. \([\mathrm{Take \ }\pi =3.142]\)