jmc

First, we sketch! We now see that $CK=BC-BK=6.$ That means $△AKC$ is a $3:4:5$ right triangle, so $AK=8.$ At this point, we can see that the area of $△ABC$ is $\frac{1}{2}\cdot AK\cdot BC=\frac{1}{2}\cdot 8\cdot 13=\boxed{52}.$