A Tale of Three Intersecting Lines
When no triangle is possible
Find out whether a triangle can be made from any three lengths.
Can any three lengths make a triangle? Try 3, 4 and 8 cm: draw the 8 cm base and swing arcs of 3 cm and 4 cm from its ends. The arcs never meet! Try 2, 3 and 6 cm: the same.
So some sets work and some don't. Must we try each one? There's a faster way, using a fact about paths.
Picture a field with a tent, a tree and a pole. You're at the tent and want to reach the tree. Which is shorter: walking straight to the tree, or walking to the pole first and then to the tree? Obviously the straight path. The direct path between two points is always shorter than going round by a third point.
Now suppose a triangle had sides 10, 15 and 30 cm (just supposing, to see what happens): BC = 10, BA = 15, AC = 30.
| Between | Direct | Round the other way |
|---|---|---|
| B and C | 10 | 15 + 30 = 45 ✓ |
| A and B | 15 | 30 + 10 = 40 ✓ |
| C and A | 30 | 10 + 15 = 25 ✗ |
From C to A, the direct path (30) would be longer than going round (25). That's impossible! So no such triangle exists, and we proved it just by reasoning, without drawing.
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