Even a shorter method ( if you want only the answer and not the procedure ) ; find the shortest side of the right-angled triangle (let it be x). If x is an odd number , the maximum radius will be (x-1)/2 units and if x is an even number , the maximum radius will be (x/2)-1 units . Hope it helps .
The way is, first draw the radius on all 3 sides it will be perpendicular to all 3 After that the right angled triangle gets divided into 3 triangles find the area of all three triangles and sum of 3 triangles will be equal to right angle triangle from there you will get an equation and after solving it you will get the radius