Question

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?

Answer 1

I know that triangle ABC and DBC are obtuse and that if the triangle is obtuse then AC^2>AB^2+BC^2

Answer 2

I plugged the numbers in and I got that the least possible integral number for triangle ABC is 11 but the least possible integral number for triangle DBC is 19 and that base is too large for triangle ABC.

Answer 3

nvm I got it

Answer 4

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.