Take a point C on the circumference of the semi-circle and join it by the end points of diameter A and B .

$\therefore$ $\angle$C = 90^$\circ$ [ by property of circle ] 

[ angle in a semi-circle is right angle ]

So, $∆$ABC is right angled triangle.

$\therefore$ Area of largest $∆$ABC = $\frac{1}{2}\times$ AB $\times$ CD 

                                           = $\frac{1}{2}$ $\times$ 2r $\times$ r 

                                           = r² sq units