How to determine if three lengths form a triangle

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

In the figure, the following inequalities hold.

a + b > c

a + c > b

b + c > a

Example 1:

Check whether it is possible to have a triangle with the given side lengths.

7 , 9 , 13

Add any two sides and see if it is greater than the other side.

The sum of 7 and 9 is 16 and 16 is greater than 13 .

The sum of 9 and 13 is 21 and 21 is greater than 7 .

The sum of 7 and 13 is 20 and 20 is greater than 9 .

This set of side lengths satisfies the Triangle Inequality Theorem.

These lengths do form a triangle.

Example 2:

Check whether the given side lengths form a triangle.

4 , 8 , 15

Check whether the sides satisfy the Triangle Inequality Theorem.

Add any two sides and see if it is greater than the other side.

The sum of 4 and 8 is 12 and 12 is less than 15 .

This set of side lengths does not satisfy Triangle Inequality Theorem.

These lengths do not form a triangle.