Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

How many

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

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: How many

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:

the 167/2 is the -1/2 i pulled out from all the even numbers in the summation, altough im a bit tired so I may have calculated wrong somewhere

Last edited by Kurre (2008-09-14 08:45:26)