Question 737271
The perimeter of a triangle can be found using:

{{{P = a + b + c}}}, where {{{a}}}, {{{b}}}, and {{{c}}} are the lengths of the sides of the triangle.

You also have the rules:
{{{a + b > c}}}
{{{a + c > b}}}
{{{b + c > a}}}

These are true for all triangles.
The most important thing to be kept in mind is the triangle inequality theorem that says that the sum of any two sides is {{{greater}}} than the third side 

If the sides do {{{not}}} {{{satisfy }}}this theorem, then the triangle is {{{not}}} {{{possible}}}

this is also good to remember:
first triangle among many triangles with perimeter {{{12}}}, the most famous is the {{{3-4-5}}} triangle, the triangle with sides {{{3}}}, {{{4}}}, and {{{5}}} (or a similar one.) Since {{{3^2 + 4^2 = 5^2}}}, we are assured by the converse of the Pythagorean theorem that the {{{3-4-5}}} triangle is right.

so, first solution is {{{4 + 3 + 5 = 12}}}


Keeping this in mind we get the following combinations:

{{{5 + 2 + 5 = 12}}}.....{{{5+2>5}}}..{{{satisfies }}}  theorem,  the triangle is  {{{possible}}}
{{{4 + 4 + 4 = 12}}}....{{{4+4>4}}}..{{{satisfies }}}  theorem,  the triangle is  {{{possible}}}

so, if {{{P = 12}}}, the possible triangles, written {{{a+ b+c}}}, are:

{{{4 + 3 + 5 = 12}}}
{{{5 + 2 + 5 = 12}}}
{{{4 + 4 + 4 = 12}}}