SOLUTION: The perimeter of a triangle is 15. The lengths of the sides are integers. If the length of one side is 6, what is the shortest possible length of another side of the triangle?

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Question 276507: The perimeter of a triangle is 15. The lengths of the sides are integers. If the length of one side is 6, what is the shortest possible length of another side of the triangle?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
The perimeter of a triangle is 15. The lengths of the sides are integers. If the length of one side is 6, what is the shortest possible length of another side of the triangle?

Suppose the sides are a, b, and 6.  Then

a%2Bb%2B6=15
a%2Bb=9

By the three triangular inequalities:

a%2Bb%3E6, a%2B6%3Eb, b%2B6%3Ec

Since a%2Bb=9, b=9-a

Substituting in the three triangular inequalities:

a%2B%289-a%29%3E6, a%2B6%3E9-a, 9-a%2B6%3Ea

9%3E6, 2a%3E3, 15%3E2a


From 2a%3E3 we have 

     a%3E3%2F2, or a%3E1.5, and since a is an integer,

     {a>=2}}}

So 2 is the smallest integer a can be.  

From 15%3E2a, we can also similarly show that 7 is the largest
integer a can be.


We could have interchanged a and b and shown the same for b, so

the smallest integer either of the sides other than the side that is 6
coul dbe is 2.

Edwin