Crossed Ladders Problem

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I tried to use the angles ($\cos$ and $\tan$) and the Intercept theorem but I don't know where is the trench in the figure. Is it just the base?

  • Could someone write down the steps by drawing each figure that's gonna be needed in your proof?

I think it usually leads to polynomial equations of degree 4:

$$x^4-6x^3-36x^2+216x-324=0$$

but I'm not sure and I need the steps for writing it in LaTex.

I think it is about $$6,326 m $$ by using Newton's method.