Triangle Inequality Theorem: Definition and Proof With Examples

In this blog, let us discuss the "Triangle Inequality Theorem".

What is Triangle Inequality Theorem?

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the third side.

Let us understand the theorem with an activity.

Draw a triangle ABC. Measure its three sides AB, BC and AC.

Triangle Inequality Theorem

AB = 3.5 cm, BC = 2.5 cm and AC = 5.5 cm
AB + BC = 3.5 cm + 2.5 cm = 6 cm,
BC + AC = 3.5 cm + 5.5 cm = 9 cm and
AC + AB = 2.5 cm + 5.5 cm = 8 cm
AB + BC > AC (6 cm > 5.5 cm)
BC + AC > AB (9 cm > 3.5 cm) and
AC + AB > BC (8 cm > 2.5 cm)   

From this activity, our statement is verified.

Triangle Inequality Theorem Example

Example: The lengths of two sides of a triangle are 6 cm and 8 cm. Between which two numbers can the length of the third side fall?

Solution: We know that the sum of two sides of a triangle is always greater than the third side. Therefore, the third side always has to be less than the sum of the two other sides.

The third side is thus less than 8 + 6 = 14 cm.

Other condition is

The side cannot be less than the difference of the two sides.
Thus the third side has to be more than 8 - 6 = 2 cm.
The length of the third side could be any length greater than 2 and less than 14 cm.