The triangle inequality tells us when three side lengths can form a real triangle. It says that the sum of any two sides must be greater than the third side. This matters because a triangle closes only when each pair of sides is long enough to meet at a vertex.
The rule is used in geometry, construction, navigation, computer graphics, and physics diagrams.
Key Facts
- Triangle inequality: a + b > c
- All three tests must be true: a + b > c, a + c > b, and b + c > a
- If a + b = c, the points lie in a straight line and do not form a triangle
- If a + b < c, the two shorter sides cannot reach each other to close the triangle
- Shortcut test: after ordering sides x ≤ y ≤ z, check only x + y > z
- For sides 4, 7, and 9: 4 + 7 = 11 and 11 > 9, so the lengths form a triangle
Vocabulary
- Triangle inequality
- The theorem stating that the sum of any two side lengths of a triangle must be greater than the third side.
- Side length
- The distance along one edge of a triangle between two vertices.
- Vertex
- A corner point where two sides of a triangle meet.
- Degenerate triangle
- A flat arrangement where the side lengths satisfy a + b = c, so the points are collinear and no true triangle is formed.
- Largest side
- The longest of the three given lengths, which is the only side that must be checked in the shortcut test.
Common Mistakes to Avoid
- Checking only whether the three lengths are positive is wrong because positive lengths can still fail to close into a triangle.
- Using a + b ≥ c is wrong because equality makes a straight line, not a triangle with area.
- Checking the two larger sides instead of the two smaller sides is unreliable because the critical test is whether the two shortest sides exceed the longest side.
- Forgetting to test all three inequalities when the sides are not ordered is wrong because any one failed inequality means no triangle can exist.
Practice Questions
- 1 Do the side lengths 5 cm, 8 cm, and 12 cm form a triangle? Show the inequality you used.
- 2 A triangle has two sides of lengths 6 m and 10 m. What whole-number values could the third side have if it is measured in meters?
- 3 Explain why three rods of lengths 3, 4, and 7 cannot make a triangle even though 3 + 4 equals the longest length.