this problem is due to Vladimir I. Arnold, who claimed that he had seen it in a standard American examination and used it as an example to criticize American math education.
such triangle does not exist. one way to see it is that the vertex of the right angle lies on a circle with the hypotenuse as a diameter, thus the corresponding altitude is at most half of the hypotenuse, which is 5 in this case.
this of course assumes Euclidean geometry. it's not difficult to embed such triangle into e.g., a 2-sphere, though it would be beyond Arnold's point as he was referring to children from age 5 to 15.