Consider that is right angled at point B.
Let and .
Let be the point on the hypotenuse in such a way that it is at a distance of from the side and at a distance of from the side .
Consider that .
Now, by Pythagoras theorem,
Since, , then,
Differentiate both sides with respect to ,
It can be observed that,
Differentiate equation (1) with respect to , we get
Thus, it can be concluded that the length of the hypotenuse is maximum when .
Using equation (2) and (3) in ,
Hence, it is proved that the maximum length of the hypotenuse is .