The smaller triangle sharing a vertex with the larger one is similar and half the area. That means its linear dimensions are √(1/2) = (√2)/2 times those of the larger triangle.
The line segment of interest has length ...
36 × (√2)/2 = 18√2
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The ratio of area of similar figures is the square of the ratio of their linear dimensions. Thus the ratio of linear dimensions is the square root of the ratio of area.
