First of all, I'm not asking for the answer to this question, but rather a hint, or a way to approach those kinds of problems.
Problem: Two sides of a triangle are a and b. What is the largest area the triangle can have? What is the shape of the triangle with the largest area?
Challenge: There is another right triangle with sides a and b. Find this triangle and its area.
The answer the book (Trigonometry by Gelfand) gave to the first part of the question, was a right triangle with an area of $ab/2$, which is okay, seems reasonable, but how would I discover such a thing? I'm stuck here.
Hint: Let $\theta$ be the angle between the two given sides. Even though $\theta$ is not specified, the area of the triangle will be $\frac12 ab\sin\theta$. This tells you that the area depends for given $a$, $b$ only on $\theta$. How does this area change as $\theta$ varies?
For the challenge problem, you'll get the same answer as before if the angle between $a$ and $b$ is $90$. If not, then either $a$ or $b$ is the hypotenuse. In that case, is there enough information to deduce the area? (For definiteness you can assume $a$ is the hypotenuse.)