Let x and y be the legs lengths.


Then the hypotenuse is  .


The "perimeter" equation is

 = 60,         (1)

which implies

 = 60 - (x+y),

 = ,

 = ,

2xy = 120*(x+y) - 3600,

xy = 60*(x+y) - 1800.      (2)


From the other side, you have this equation expressing the area of the right-angled triangle 

 = ,

which gives

xy = .           (3)


In this equation, replace    in the right side by  [60 - (x+y)], based on  (2).


You will get instead of (3)

xy = 12*(60 - (x+y)).       (4)


Now compare equations (2) and (4).  Their left sides are identical; hence, right sides are equal:

60*(x+y) - 1800 = 12*(60 - (x+y)).


It implies  60*(x+y) + 12*(x+y) = 12*60 + 1800,  or 72*(x+y) = 2520,  and then  x+y =  = 35.


Thus we found that the sum of the leg lengths is 35 inches.  


Then the hypotenuse is  60 - 35 = 25 inches  (from the condition on the perimeter),


and the area of the triangle, which is under the question, is   = 25*6 = 150 square inches.