SOLUTION: Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is t

0.  Make a sketch to follow my arguments.

    Let O be the center of the circle located on the leg AB, and 

        P be the tangent point lying on the hypotenuse AC.



1.  Two right-angled triangles are similar: triangle ABC and triangle APO.



2.  Regarding the triangle ABC, notice that its sides are 5 (the leg AB),  (the hypotenuse AC) and  =  = 6 (the other leg BC).


3.  Let r be the unknown radius of the circle.



4.  Use two proportions that follow the similarity of the triangles:

     = ,

    which after substituting the data takes the form

     = .