The answer is simple: the triangle can't have an area because there is no such triangle.

Why? That's because any right triangle (I made a typo in the question, I apologize for that) can be inscribed in a circle with the hypotenuse as its diameter.

So, if the triangle is inscribed in a circle with a diameter of 10 cm (the lenght of the hypotenuse), then the max lenght of the altitude to the hypotenuse is 5 cm; 6 cm is bigger than 5 cm, so the triangle can not exist.

So, the ONLY correct answer is: there is NO such triangle. Anyone who gives me a value is wrong.