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one want to find the length of the two hypotenuses of the figure

Here is what I did. I looked at the small first with hypotenus unknown and adjacent=1 and the other cathetus unknown. Then i wanted to find max value of the smallest hypotenus.

[tex]tan\theta=\frac{1}{a}[/tex]

[tex]a=\frac{1}{tan\theta}[/tex] Smallest hypotenus=sh

[tex]sh^2=1+\frac{1}{tan^2\theta}[/tex]

i took derivative to find max value

[tex]sh=\sqrt{1+cot^2\theta}[/tex]

[tex]\frac{d}{dx}sh=\frac{1}{\sqrt{1+cot^2\theta}} \cdot \frac{coth\theta}{sin^2\theta}[/tex]

but this is equal to zero when [tex]\theta=\frac{\pi}{2}[/tex] whoch gives sh=1

which is obviously wrong since adjacent cathetus=1

What is wrong?

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# Length of stick by derivation

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