9. The radii of two concentric circles are in the ratio $1: 3$. $A C$ is a diameter of the larger circle. $B C$ is a chord of the larger circle and is tangent to the smaller circle. $A B$ has length 140 .
What is the radius of the larger circle?
Solution
210

Let $O$ be the centre of the circles, let $T$ be the point where the chord $B C$ meets the smaller circle and let $r$ be the radius of the smaller circle.
$\angle C T O=90^{\circ}$ by the tangent-radius theorem.
$\angle C B A=90^{\circ}$ by the angle in a semicircle theorem.
Therefore, $\triangle C T O$ is similar to $\triangle C B A$ since both are right-angled and they share $\angle O C T$.
Now, $O T: A B=C O: C A=1: 2$.
However, $A B=140$, so $r=70$ and the radius of the larger circle is $3 \times 70=210$.