A chord PQ of a circle is 16 cm long. A diameter AB of the same circle crosses PQ at the midpoint M of that chord, and the centre O is 6 cm from M. What is the length of the diameter AB?

M is the midpoint of PQ, so PM = MQ = 8 cm. AB and PQ are two chords crossing at M, so AM × MB = PM × MQ = 64. Writing the radius as $r$, the two parts of the diameter are AM = $r-6$ and MB = $r+6$, so $(r-6)(r+6)=r^2-36=64$. Then $r^2=100,r=10$ cm, and the diameter is 20 cm. The value 10 cm is the radius, not the diameter.