I have this right triangle 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".
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 $0 \leq x \leq \frac{\pi}{2}$.