Question 1088328
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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), {{{sqrt(61)}}} (the hypotenuse AC) and {{{sqrt(61 - 5^2)}}} = {{{sqrt(36)}}} = 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:

    {{{abs(OP)/abs(AO)}}} = {{{abs(BC)/abs(AC)}}},

    which after substituting the data takes the form

    {{{r/(5-r)}}} = {{{6/sqrt(61)}}}.
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