A semi-circle is drawn outwardly on chord AB of the circle with centre $O$   and unit r adius.  The perpendicular from $O$   to AB meets the semi-circle on AB at $C$  .

if  $C\text{'}$  is any other point on the semi-circle, then  $OC<OC\text{'}$

if $C\text{'}$  is any other point on the semi-circle, then $OC>OC\text{'}$ 

$AB=\sqrt{2}$ , when OC has maximum length.

maximum length of OC is 2