# Combinatorics in Geometry

**Discrete Mathematics**Level 4

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 \(y\)?