# Radius of an Inscribed Semicircle

Geometry Level 3

$$ABC$$ is a right angled triangle with $$\angle ABC = 90^\circ$$ and side lengths $$AB = 36$$ and $$BC = 27$$. A semicircle is inscribed in $$ABC$$, such that the diameter is on $$AC$$ and it is tangent to $$AB$$ and $$BC$$. If the radius of the semicircle is an improper fraction of the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, what is the value of $$a + b$$?

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