Step 1: Set up the optimization problem.
Legs $ a,b $; hypotenuse $ h $; constraint: $ a^2+b^2=h^2 $; maximize area $ S=\frac{ab}{2} $.
Step 2: Apply AM-GM to $ a^2 $ and $ b^2 $.
\[ \frac{a^2+b^2}{2}\geq\sqrt{a^2 b^2}=ab \implies ab\leq\frac{h^2}{2} \]
Step 3: Identify when equality holds.
AM-GM equality holds iff $ a^2=b^2 $, i.e., $ a=b $. The triangle is isosceles right-angled.
Step 4: Compute the maximum area.
\[ S_{\max} = \frac{1}{2}(ab)_{\max} = \frac{1}{2}\cdot\frac{h^2}{2} = \frac{h^2}{4} \]
Step 5: Verify with $ h=2 $.
$ a=b=\sqrt{2} $, Area $ =\frac{1}{2}\cdot\sqrt{2}\cdot\sqrt{2}=1=\frac{4}{4} $. Confirmed.
Step 6: State the maximum area.
\[ \boxed{\dfrac{h^2}{4}} \]