Find the minimal length of a right triangle with altitude 1

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

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