Find the area

The median on the hypotenuse of a right triangle equals one-half the hypotenuse. (Special case of Apollonius's theorem, or of the parallelogram law/Converse of the Theorem of Thales. I really don't know the name for this theorem.)

Thus,

$\overline{AB} = 13$ .

Using the Pythagorean theorem, we find

$\overline{BO} = \sqrt{(\overline{AB})^2 - (\overline{AO})^2} = \sqrt{13^2 - 12^2} = \sqrt{25} = 5$ .

Therefore, the area of the triangle is

$= \frac {1}{2} \cdot \overline{AO} \cdot \overline{BO} = \frac {1}{2} (12) (5) = 30$ .