Find the area of triangle ABC

Construct an altitude h of the triangle:

Since $\angle B = \sin^{-1}\frac{24}{25}$ , let $h=24x$ . Then $c = 25x$ and, since $\angle C = \sin^{-1}\frac{12}{13}$ , $b = 26x$ . As both 7, 24, 25 and 10, 24, 26 are pythagorean triples, $17 = a = 7x + 10x = 17x$ . Thus $x=1$ and the area of $\triangle ABC$ is $\frac{17\cdot24}{2}=204$ .