G3.4 In Figure $4, A D, B C$ and $C D$ are tangents to the circle with centre at $O$ and diameter $A B=12$. If $A D=4$, find the value of $B C$.
Suppose $C D$ touches the circle at $E$. Let $B C=x$.
$D E=4$ and $C E=x$ (tangent from ext. point)
From $D$, draw a line segment $D F / / A B$, cutting $B C$ at $F$.
$$
\begin{array}{l}
\angle D A B=\angle A B C=90^{\circ} \text { (tangent } \perp \text { radius) } \\
\angle D F C=90^{\circ} \text { (corr. } \angle \text { s AB //DC) }
\end{array}
$$
$\therefore A B F D$ is a rectangle.
$$
\begin{array}{l}
D F=12, B F=4 \text { (opp. sides of rectangle) } \\
C F=x-4
\end{array}
$$

In $\triangle C D F,(x-4)^{2}+12^{2}=(x+4)^{2}$ (Pythagoras' theorem)
$$
\begin{array}{l}
x^{2}-8 x+16+144=x^{2}+8 x+16 \\
144=16 x \\
B C=x=9
\end{array}
$$