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## 'As Long as Possible' printed from http://nrich.maths.org/

$14 \; \text{cm}$. The length of $AD$ must be less than $15 \;
\text{cm}$, since $15 \; \text{cm}$ would be its length if all four
points lay in a straight line. However, by making angles $ABC$ and
$BCD$ close to $180^{\circ}$, $AD$ can be made close to $15 \;
\text{cm}$ in length.

As the length of $AD$ is a whole number of centimetres, its maximum
value, therefore is $14\; \text{cm}$

*This problem is taken from the UKMT Mathematical Challenges.*