Question

x and y are whole numbers, 10<x<14, and 8<y<14.
The sides of a triangle are 5 cm, x cm, and y cm.
List possible values for x and y.

Answer 1

since any side is less than the sum of the other two sides,

x+y > 5 (duh)
x+5 > y
y+5 > x

lots of possible solutions:

5,11,9
5,13,13
and more

Answer 2

values of x could be 11, 12, or 13
values of y could be 9, 10, 11, 12 or 13

now overthinking the problem:

To have a triangle:
1. x+y > 5
2. x+5 > y ----> x-y > -5 or y-x < 5
3. y+5 > x ----> y-x > -5

from #2 and #2 we can see that
-5 < y-x < 5

so make 3 columns
5_x__y_(y-x)

5 11 9
5 11 10
5 11 11
5 11 12
5 11 13

5 12 9
5 12 10
..
5 12 13

5 13 9
..
5 13 13

in short, the x could be 11, 12 or 13
the y could be 9, 10, 11, 12, or 13

so the number of triangles for sides labeled x , y and 5 is 3(5) or 15
However, triangles 5 , 12, 13 and 5, 13, 12 are the same triangle
there are 3 such duplicated pairs, so
the actual number of triangles possible is 15-3 = 12