Cake cutting with shortest cut

Geometry Level 5

A triangle has side lengths 13, 14 and 15. A line segment divides this triangle into two polygons, one of which has an area twice that of the other. Find the smallest possible length of the segment.

If you get your answer as \(a \sqrt{b}\), where \((a,b) \in \mathbb{N}^2\) and \(b\) is not divisible by the square of any prime, submit your answer as \(a+b.\)

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