$\triangle abc$ and $\triangle dbc$ share $bc$. $ab = 5\ \text{cm}$, $ac = 12\ \text{cm}$, $dc = 8\ \text{cm}$, and $bd = 20\ \text{cm}$. what is the least possible integral number of centimeters in $bc$?

What terms govern the length of this side?The basic rule of the triangle.First side length must be less than the sum of the other two sides.So to find X we must take the largest side of the triangles and compare them with amounts from other sides.

5+x>12
8+x>20    and it's system

x>7
x>12
general solution is x>12
The least possible integral is 13.

PS: It's may be yet 12, but in this case, triangle  BCD become segment.