Triangle ABC has a right angle at B, with AB = 15 cm and BC = 20 cm…

Step 1. The hypotenuse. $AC^2 = 15^2 + 20^2 = 625$, so $AC = 25\ \text{cm}$.

Step 2. The diameter. $\angle ABC = 90^\circ$, so $AC$ is a diameter of the circle. The diameter is $25\ \text{cm}$ and the radius is $12.5\ \text{cm}$.

Step 3. The chord $AB$. Let $N$ be the foot of the perpendicular from the centre $O$ to $AB$. It bisects $AB$, so $AN = 7.5\ \text{cm}$.

Step 4. The distance. $ON^2 = 12.5^2 - 7.5^2 = 156.25 - 56.25 = 100$, so $ON = 10\ \text{cm}$.