Triangle ABC and triangle DBC share BC. AB=5 cm AC=12 cm DC=8 cm and BD= 20 cm. What is the least possible integral number of centimeters in BC?
By the triangle inequality on triangle $ABC$, $BC > AC - AB = 12 - 5 = 7$, and by the triangle inequality on triangle $BCD$, $BC > BD - CD = 20 - 8 = 12$. Hence, $BC$ must be at least $\boxed{13}$ centimeters. (And it is easy to verify that it is possible for $BC$ to be 13 centimeters.
To find the least possible integral number of centimeters in BC, we need to apply the Triangle Inequality Theorem.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
In this case, triangle ABC has sides AB = 5 cm and AC = 12 cm, while triangle DBC has sides DC = 8 cm and BD = 20 cm.
Let's consider the side BC. According to the Triangle Inequality Theorem, the sum of BC and AB must be greater than AC, and the sum of BC and DC must be greater than BD.
BC + AB > AC
BC + 5 > 12
BC > 12 - 5
BC > 7
BC + DC > BD
BC + 8 > 20
BC > 20 - 8
BC > 12
Since we want to find the least possible integral number of centimeters in BC, we round up to the next whole number.
BC must be greater than 12 and 7, so the smallest integral value that satisfies both conditions is BC = 13 cm.