Can Three Sticks Meet?

Part of the Triangles section of Coddy's Geometry journey. Lesson 7 of 71.

Not every three lengths make a triangle. Try sticks of 3, 4 and 8. Lay the stick of 8 flat. Attach the stick of 3 to one end of it and the stick of 4 to the other end. However you turn the two short sticks, their free ends never meet. The two short sticks together are only 7 long, and the ends of the 8 are 8 apart.

$A stick of length 8 with sticks of 3 and 4 raised from its ends, leaving a gap between their tips$

Sticks of 3 and 4 are too short to meet across a stick of 8

The triangle inequality is the rule behind this. In every triangle, any two sides added together are longer than the third side. To test three lengths, add the two shorter ones and compare the sum with the longest. If the two shorter sides pass this test, the other two pairs pass too, because each of those pairs includes the longest side. For 3, 4 and 5, 3 + 4 > 5, so these lengths make a triangle. For 3, 4 and 8, 3 + 4 < 8, so these lengths do not.