Question 702624
Right triangle shown below, hypotenuse unlabeled, and legs x and y lengths.
Area was given as 4 [square units]:
{{{(1/2)*x*y=4}}}


The sum of the legs' lengths given as 9 units:
{{{x+y=9}}}


Now we have two equations and two unknowns, x and y.  
From the area equation, we obtain {{{x=8/y}}}, which we directly substitute into the length equation to obtain {{{(8/y)+y=9}}}.  Clearing the fraction yields 
{{{y^2-9*y+8=0}}}
.
...General solution to quadratic formula will give y = 1 or 8.  This is just fine.  Checking each using either equation from our system, gives x = 8 or 1.


ANSWER: The length of the legs are 1 and 8.
(please scroll down to see picture.)


{{{drawing( 300, 320, 0, 10, 0, 10,
            line( 1,1,9,1),
            line( 9,1,9,2),
            line( 9,2,1,1),
            locate(5,0.5,X),
            locate(9.5,1.5,Y) 
)}}}