Question

The number of different non congruent triangles with integer side and perimeter 15 is

Answer 1

x + y + z = 15

x + y > z
x + z > y
y + z > x

From
x + y + z = 15
x + y = 15 - z

since x + y > z,
15 - z > z
15 > 2z
7.5 > z

agn
x + y + z = 15
x + z = 15 - y

since x + z > y,
15 - y > y
15 > 2y
7.5 > y

agn
x + y + z = 15
y + z = 15 - x

since y + z > x,
15 - x > x
15 > 2x
7.5 > x

So all the sides are 7 or less

sure they are non-congruent, 

we'll require that perimeter=15

x = 1 y = 7 z = 7
x = 2 y = 6 z = 7
x = 3 y = 5 z = 7
x = 3 y = 6 z = 6
x = 4 y = 4 z = 7
x = 4 y = 5 z = 6
x = 5 y = 5 z = 5

So there are 7

Hope it helps...☺