SOLUTION: A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What

Question 266509: A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What is the area of the triangle?
Found 2 solutions by Edwin McCravy, vksarvepalli:
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!
A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What is the area of the triangle?

It's a matter of solving this system of equations and substituting in the right triangle area formula Clear fractions in the third equation: Solve the first equation for Therefore Substituting in Solve for c Substitute that and in the Pythagorean equation Cancel the 's It isn't necessary to find c but we'll do it anyway so we can check if we like. Substitute that in But all we want is the area, so we substitute and A = 134,670 square units. It was not necessary to tell us that the sides are integers. Edwin

Answer by vksarvepalli(154) (Show Source): You can put this solution on YOUR website!
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
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