Hello, cutiepie!
A rectangle inscribed inside of an isosceles right triangle with hypotenuse 2.
What is the largest area a the rectangle can have, and what are its dimensions? I'll assume the diagram looks like this: Code:

1*
*  *
*  * P
*   +   o(x,y)
*    *
*   y *
*    *
  *    +   +   +    *  
1 x  x 1
Point $\displaystyle P$ is on the line: $\displaystyle y \:=\:1x$ .[1]
The area of the rectangle is: .$\displaystyle A \;=\;2xy$ .[2]
Substitute [1] into [2]: .$\displaystyle A \;=\;x(1x) \quad\Rightarrow\quad A \;=\;xx^2$
Maximize: .$\displaystyle A' \;=\;1  2x \:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:\tfrac{1}{2}}$
Substitute into [1]: .$\displaystyle y \;=\;1\tfrac{1}{2} \quad\Rightarrow\quad\boxed{ y \:=\:\tfrac{1}{2}}$
Therefore, its dimensions are: .$\displaystyle \begin{array}{ccccc}\text{Length} &=&2x &= & 1\\ \text{Width} &=& y &=& \tfrac{1}{2} \end{array}$
. . and its maximum area is: .$\displaystyle A \;=\;(1)\left(\tfrac{1}{2}\right) \;=\;\tfrac{1}{2}$