Çözüldü Right Triangle - Trigonometry

On a right triangle KLM with |KL| = 5 units and |KM| = 13 units, if a point P is chosen on the edge [LM] by folding the vertex L over the line [KP] causing the new point of L to overlap the point L' on the hypotenuse [KM], what is the tangent value of the angle ∡MKP = x?

Based on the symmetry of the deltoid KLPL';
cos(2x) = 5 / 13 = 2·[ cos(x) ]^2 - 1 ⇒ cos(x) = 3 / √13 ⇒ tan(x) = [ (√13)^2 - 3^2 ]^0,5 / 3 = 2 / 3.

Original Turkish version of the problem with another solution using an unnecessary variable k:

https://i.ibb.co/xShgzxsZ/gen.png
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