# A right angled triangle has integral sides(the length of sides are positive integers). if the smallest side is 2003, then find the value of the perimeter of the triangle.

1
by sumaira997
is it home work?

2014-10-27T14:19:52+05:30

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Let a = 2003.           b and c are the other leg and the hypotenuse.
Let c - b = x            all b, c, and x are integers

c² - b² = 2003 * 2003
(c - b) (c +b) =   x ( 2 c - x) = 2003 * 2003

2003 is a prime number.   Factors on LHS must be equal to factors on RHS.

(I) x = 2003   and  2c - x = 2003    or   (2)  x = 1   and  2 c - x  = 2003²

(1)  c = 2003   b = 0  a = 2003  this is not a triangle.

2)  c = ( 2003² + 1 )/2 = 2006005               b = 2006004      a = 2003

Verify   2003² + 2006004² = 2006005²

Perimeter = 4014012