The Triangle Inequality

Triangle Inequality Theorem

The sum of the lengths of any two sides of a Triangle must be greater than the length of the third side.

For a Triangle with sides a, b, c:

• a + b > c
• a + c > b
• b + c > a

Using the Triangle Inequality

To check if three lengths can form a Triangle, verify all three inequalities. In practice, you only need to check that the sum of the two shortest sides is greater than the longest side.

Example 1: Can sides 3, 4, 8 form a Triangle?

Check: 3 + 4 = 7 < 8. No, these cannot form a Triangle.

Example 2: Can sides 5, 7, 10 form a Triangle?

Check: 5 + 7 = 12 > 10. ✓ Also 5 + 10 > 7 ✓ and 7 + 10 > 5 ✓. Yes.

Finding the Range of the Third Side

If two sides have lengths a and b (where a < b), the third side c must satisfy:

b − a < c < a + b

Example: If two sides are 5 and 11, then the third side must be between 11 − 5 = 6 and 5 + 11 = 16: 6 < c < 16.