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I have this right triangle here.

enter image description here

The question says: "Suppose we have a right triangle $ABC$, where the right angle is at $C$. Draw the altitude from vertex C to hypotenuse $AB$. If the length of this altitude is $1$ cm, what is the minimal length of the hypotenuse?"

Can someone help me set up the optimization problem? I'm honestly having a hard time setting this up and I'm genuinely stuck. I am given a hint which says "denote by $x$ one of the acute angles".

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Let $D$ be the point of intersection between the drawn altitude an AB. Then,

$$AB=AD+DB$$

From trigonometry,

$$AD=\cot x$$

From the geometry of the problem, angle $DCB$ is also $x$, so:

$$DB=\tan x$$

All that is left is to minimize,

$$AB=\cot x+\tan x$$

Subject to $0 \leq x \leq \frac{\pi}{2}$.

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  • $\begingroup$ Thank you. I figured out the rest of it :) . $\endgroup$ – Future Math person Nov 20 '18 at 2:37

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