Isosceles right triangle ABC encloses a semicircle of area 2π. The circle has its center O on hypotenuse AB and is tangent to sides AC and BC. What is the area of triangle ABC?[asy]defaultpen(linewidth(0.8));pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
draw(circle(o, 2));
clip(a--b--c--cycle);
draw(a--b--c--cycle);
dot(o);
label("C", c, NW);
label("A", a, NE);
label("O", o, SE);
label("B", b, SW);[/asy]<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>3π<spanclass=′latex−bold′>(D)</span>10<spanclass=′latex−bold′>(E)</span>4π