The triangular inequality

Can a triangle always be constructed?

No! Given any three lengths, it is not always possible to construct a triangle. There is a rule known as the triangle inequality.

The rule

For three line segments of lengths a, b and c to form a triangle, the sum of the two shorter sides must always be strictly greater than the longest side:

a + b > c (where c is the longest of the three).

An example that works

With side lengths of 3 cm, 4 cm and 5 cm: the longest side is 5. We check that 3 + 4 > 5, which is 7 > 5. This is true, so this triangle exists.

Example that does not work

With side lengths of 2 cm, 3 cm and 6 cm: the longest side is 6. We check that 2 + 3 > 6, which is 5 > 6. This is false! These three line segments cannot form a triangle: when placed end to end, the two shorter sides do not even meet to close the figure.

Borderline case

If a + b = c exactly, the three points are collinear: the figure is flat; it is not a true triangle.

Common pitfall

Many pupils only try a single random addition. You must always compare the sum of the two SMALLEST sides with the LARGEST side: this is the most difficult combination to satisfy, and therefore the only one that really matters.