Could you find the length of AB?

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[mohanramachandran]

Very much interesting. Simple solution.
Thank you.

[satyanarayanmohanty]

Applying trigonometry. a/y=tan2σ and x/a=tanσ. Where a is the side of square. Therefore we get a^2=x^2+2xy. Now AB=x+y.

[harikatragadda]

Rotate a copy of the square 90°clockwise around the corner A.
We now get an Isosceles triangle with angles (90- θ, 90- θ, 2θ).
This gives AB= X+Y.