Question 975697
x and y for the legs, h for the hypotenuse.


A, for area.


{{{system(x+y+h=200,A=(xy)/2,x^2+y^2=h^2)}}}


{{{h=200-x-y}}}
{{{h=200-(x+y)}}}
-
{{{h^2=(200-(x+y))^2}}}
{{{h^2=40000-400(x+y)+x^2+2xy+y^2}}}
All that just from the first, "200" perimeter equation.  This means the two formulas for h^2 can be equated to eliminate h.


{{{x^2+y^2=40000-400(x+y)+x^2+2xy+y^2}}}
Both sides have x^2+y^2...
{{{0=40000-400(x+y)+2xy}}}-----can this be solved either for x or for y?

{{{40000-400x-400y+2xy=0}}}
{{{200000-200x-200y+xy=0}}}
{{{-200y+xy=200x-20000}}}
{{{xy-200y=200x-20000}}}
{{{(x-200)y=200x-20000}}}
{{{y=(200x-20000)/(x-200)}}}


Use THAT formula of y in the area equation:
{{{A=(xy)/2}}}
{{{highlight_green(A=x((200x-20000)/(x-200)))}}}-----formula for AREA, in terms of only x, according the the description's conditions.


{{{graph(300,300,-10,200,-10,200,x((200x-20000)/(x-200)))}}}----Not displaying; not sure why.


You can use Google search engine, input y=x((200x-20000)/(x-200)) and enter, and you can see the graph, and can zoom in and shift the position of the framing to see a good local maximum.  "y" as used in the graphing tool will stand for AREA, not the y variable of the dimension as in the problem description.