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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