Find the area of the triangular region ABC below
Solution 1:
Realize that $AT$ and $TC$ are also tangent segments, you can compute their lengths same way you found $AC$.
Next, extend $OO'$ to meet $AC$ in $D$. Now triangles $BTD$ and $OAD$ can be shown to be similar with ratio $1/2$. Hint : Join $O'C$ and find some $3:4:5$ triangles.
This will yield $BT=OA/2=18$.