Law of Cosines Method: Trigonometry Rules!!!
Let the center of the semicircle be P, and the center of the small circle be Q. Additionally, let the midpoint of AB = M, and the midpoint of BC = N. I'm about to spit out constructions, so try to follow along:
We can construct a triangle MQP, with side lengths MQ = 2 + r, QP = 5 - r, and PM = 5 - 2 = 3.
Additionally, triangle NQP, with side lengths NQ = 3 + r, QP = 5 - r, and PN = 5 - 3 = 2.
Using the Law of Cosines (which I expect that you should know already) on NQP:
(3+r)2=(5−r)2+22−2(5−r)(2)cos(QPN)=r2+6r+9=r2−10r+29−(20−4r)cos(QPN)
16r−204r−20=cos(QPN)=4r−5r−5
Using the Law of Cosines on MQP: (and substitution)
(2+r)2=32+(5−r)2−2(3)(5−r)cos(QPM)=r2+4r+4=r2−10r+34+(30−6r)(4r−5)(r−5)
14r−30=−(6r−30)(4r−5)(r−5)=−6(4r−5)=−24r+30
Note that cos(QPM) = -cos(QPN)
Solving this equation, you get 38r = 60, and r = 30/19.