A is at a reference point. B is 5 km north of A. C is 12 km east of A.

Using the Pythagorean theorem:

$BC^2 = AB^2 + AC^2$

Substituting the values:

$BC^2 = 5^2 + 12^2$

$BC^2 = 25 + 144$

$BC^2 = 169$

$BC = \sqrt{169}$

$BC = 13$ km

Therefore, the shortest distance between B and C is 13 km.