The only information provided is the lengths of the sides of the triangle
Question: Find the radius of the circle. Ans:2cm
$\displaystyle Angle ACB=sin^-1\frac{5}{13}
$$\displaystyle =22.6198$
Let length of radius be $\displaystyle x$ and point of centre of circle be $\displaystyle O$
$\displaystyle Angle OCA=\frac{22.6198}{2} $
$\displaystyle =11.3099 $(external point of tangent of circle)
$\displaystyle tan11.3099=\frac{x}{13}$
$\displaystyle x=13tan11.3099$
$\displaystyle =2.6cm$
The circle does touch the triangle. Else, it can't be solved.
Lets call the point of intersection (for the one above ) be P and the one below, Q, and the one by the side, R.
AP=AR=x and RB=BQ=y
CP=13-x and CQ=12-y
x+y=5
13-x=12-y
Solving the simultaneous equation yields x=3, y=2.
angle ACB=22.62
angle COP=11.3
tan 11.3 = radius / 10
radius =2