Question 666171: How many different isosceles triangles have all interger side lengths an a perimeter of 30? List them. Explain what you need to check for to ensure that your triangle is correct.
Answer by swincher4391(1107)   (Show Source):
You can put this solution on YOUR website! Let a,b,c be sides of a triangle such that:
a=b or b=c or a=c but not a=b=c (that would be equilateral)
In addition we have that a+b>c or a+c > b or b+c > a (by triangle inequality)
Notice that if two sides are equal, then if the other side is less, it's always a triangle. The only way we don't have a triangle is if we have a,b be our equal sides and c be the longest side and c>2a. So 2a < c.
Let a=b.
c must be less than 15.
So a must be 8 or 9.
a can't be 10 because then it is equilateral.
a can be 11 through 14 since c will be 2 through 8.
So a = 8,9,11,12,13,14.
So we have 6 isosceles triangles with perimeter 30.
8,8,14
9,9,12
11,11,8
12,12,6
13,13,4
14,14,2
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