Combinatorics in Geometry

In triangle \(\Delta ABC\), \( \overline {AB} = 12 \), and \( \overline {AC} = 5\). Given that \( \overline{BC} \) is chosen uniformly in the interval of permissible values such that \(\Delta ABC\) is a non-degenerate triangle .

The probability that \(\Delta ABC\) is an acute triangle can be expressed in the form \( \large \frac{a - \sqrt{b}}{c} \) where \(a\), \(b\) and \(c\) are coprime positive integers. Determine \(a + b + c\).

Follow up question: Could you generalize this probability for any triangle with two known sides \(x\), and \(y\)?

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