How do you find the altitudes of a right triangle with the three sides?

Answer 1

If you just want the area, use Hero's formula:
Let
p = (a+b+c)/2 = half-perimeter
Area = √(p(p-a)(p-b)(p-c))

If it's really the altitude you need, then find each angle using the cosine law:
cos(A)=(b²+c²-a²)/(2bc)
where a,b,c are lengths of sides opposite to the respective angles A, B and C. Altitudes are AD, BE, CF.
Altitude CF can be found by b*sin(A), etc.