Question 266509
let the sides be a,b & c with a being the shortest and c the hypotenuse(longest)

the are of the triangle will be 1/2*a*b  (half*base*height)

given  

The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides

so a=1/5 *(b+c)

=> 5a=b+c      ------------------ 1

and perimeter = 2010 units

so a+b+c = 2010

but from 1 

we get a+5a=2010

so 6a=2010

so a=335

and b+c=5a=1675

now in a right triangle fro Pythagoras theorem we have

c^2 = a^2+b^2

=> c^2-b^2=a^2
 
=> c^2-b^2=335*335

=> (c-b)(c+b)=335*335

but c+b = 1675

so (c-b)*1675=335*335

so c-b = 67
   c+b= 1675

subtracting the above two we get 

2b=1608  so b= 804

thus area=1/2 * 335 * 804

therefore  area of the triangle=134670 sq. units