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?
<pre><font size = 4 color = "indigo"><b>
Suppose the sides are a, b, and 6.  Then

{{{a+b+6=15}}}
{{{a+b=9}}}

By the three triangular inequalities:

{{{a+b>6}}}, {{{a+6>b}}}, {{{b+6>c}}}

Since {{{a+b=9}}}, {{{b=9-a}}}

Substituting in the three triangular inequalities:

{{{a+(9-a)>6}}}, {{{a+6>9-a}}}, {{{9-a+6>a}}}

{{{9>6}}}, {{{2a>3}}}, {{{15>2a}}}


From {{{2a>3}}} we have 

     {{{a>3/2}}}, or {{{a>1.5}}}, and since a is an integer,

     {a>=2}}}

So 2 is the smallest integer a can be.  

From {{{15>2a}}}, 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</pre>