OM is the perpendicular from the centre O of a circle to the chord AB…

Step 1. The equal parts.

1. $\angle OMA = \angle OMB = 90^\circ$.
2. $OA = OB$, because they are radii.
3. $OM = OM$, the common side.

Step 2. The rule. $\triangle OMA \cong \triangle OMB$ by RHS. So $AM = MB$ by CPCT, and $M$ is the middle point of $AB$.

Step 3. The radius. $AM = 9\ \text{cm}$. In triangle $OMA$, $OA^2 = 12^2 + 9^2 = 225$, so $OA = 15\ \text{cm}$.