The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
AB + BC > AC
BC + AC > AB
AC + AB > BC
Why is this important?
If any of these three inequalities is not true, you do not have a triangle.
For example, let 2,3 and 6 be the given lengths. Let’s check if they can form a triangle by applying the triangle inequality:
2 + 6 > 3 ?
3 + 6 > 2 ?
2 + 3 > 6 ?
The last inequality is not true and therefore the three given lengths cannot form a triangle. We can graph the given lengths to better understand what this means.
Conclusion: Not any three lengths can be lengths of sides of a triangle. They need to satisfy the triangle inequality.
Note: that you don’t always have to check all three inequalities; it is enough to check if the sum of the two shorter sides is greater then the largest side of a triangle.
Could a triangle have side lengths of 1, 5 and 9?
Let’s use the triangle inequality theorem and examine all 3 combinations of the sides:
1 + 9 > 5 ?
1 + 5 > 9 ?
We see that 1+5 is not greater than 9, so these lengths do not satisfy the theorem and they couldn’t form a triangle. There’s no need to check the third inequality because we already have one that doesn’t satisfy the theorem and that is enough to get the conclusion.
Could a triangle have side lengths of 5, 9 and 10?
We’ll use the shortcut – we can check if the sum of the two shorter sides is greater than the largest side. If so, a triangle could have side lengths of 5, 9 and 10.
5 +9 > 10 ?
This is true, so yes, a triangle could have these side lengths.
If the sum of two sides is equal to the third side, they still don’t form a triangle. Instead, they form a straight line.
For example, the side lengths of 2, 4 and 6:
2 + 4 > 6 ?
This is incorrect, so the three lengths do not form a triangle.