Random Point in a Hexagon

Discrete Mathematics Level 5

A point \(P\) is uniformly chosen inside a regular hexagon of side length \(3\). For each side of the hexagon a line is drawn from \(P\) to the point on that side which is closest to \(P\). The probability that the sum of the lengths of these segments is less than or equal to \(9\sqrt{3}\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

The side of the hexagon is a line segment, not a line.

Note that the 6 closest points are always distinct, hence we will have 6 distinct line segments.


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