In a circle of radius 5 cm, AB and AC are two chords, each 6 cm long. Find the length of BC.

The centre O lies on the line from A that bisects BC at right angles at M. With AM = $h$ and BM = $x:x^2+h^2=36$ and $x^2+(5-h{)}^2=25$. Subtracting gives $25-10h=-11,h=3.6$, so $x=\sqrt{36-12.96}=4.8$ and BC = 9.6 cm.