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.
kvnmurty:
is it home work?
Answers
Answered by
3
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
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
Similar questions