How many triangles are there with integer sides and perimeter 2007? How many of these are equilateral? How many are isosceles? How many are scalene
Let the triangle have sides a,b,c, with a≥b≥c. We want to solve the equation a+b+c=2007 in positive integers.
we have first that 669≤a≤1003, because if a is less that 2007/3=669, then b or c must be greater than a, if a is larger than 1003, then a>b+c contradicting the triangle inequality.
for a=669, then b=c=669 we have the equilateral triangle. Assume that a is greater than 669. If a is odd, then we start with the triangle b=c=(2007-a)/2. Now we may increase b and decrease c up to b=a, ie for any odd a, there are a-(2007-a)/2+1=3a/2-2005/2 triangles. For even a, we start with b=(2008-a)/2, c=(2006-a)/2. Now there are a-b+1=a-(2008-a)/2+1=(3a/2-2005/2)-1/2 different triangles. Summing these two yields number of triangles to:
Last edited by Kurre (2008-09-14 08:45:26)