Triangle Inequality Theorem

We know that a triangle has three sides. But, have you ever thought what is necessary for the three line segments to form a triangle. Is it possible to make a triangle with any three line segments?

In the given figure, line segments 6, 8, and 10 units form a triangle. What about if the line segments change to 6, 8, and 17 units?

We would not be able to create the triangle as shown by an incomplete triangle. This proves that we cannot create a triangle from any combination of three line segments. This relationship is explained using the triangle inequality theorem.

What is the Triangle Inequality Theorem

The above theorem describes the relationship between the three sides of a triangle. It tells us that for 3 line segments to form a triangle, it is always true that none of the 3 line segments is greater than the lengths of the other two line segments combined.

Let us take our initial example. We could make a triangle with line segments having lengths 6, 8, and 10 units. This is because those line segments satisfy the triangle inequality theorem.

6 + 8 = 14 and 10 < 14

8 + 10 = 18 and 6 < 18

6 + 10 = 16 and 8 < 16

Here, we see that none of the line segments are longer that the sum of the other two line segments.

In contrast, if we consider the line segments of lengths 6, 8, and 17 units, we find that the line segment measuring 17 units is longer than the length of the other two line segments combined.

This proves that we cannot make a triangle with these three line segments. Thus, they don’t satisfy the triangle inequality theorem.

The inequality theorem is applicable to all types of triangles such as scalene, isosceles, and equilateral.

Triangle Inequality Theorem Proof

To prove: |YZ| <|XY| + |XZ|

Proof:

The side XZ is extended to a point W such that XW=XY as shown in the given figure.

Hence proved that, the sum of the lengths of any two sides of a triangle is greater than the third side.

Let us solve some problems involving the above theorem to understand the concept better.

Solved Problems

If 6cm, 12cm and 4cm are the measures of three lines segment. Can it be used to draw a triangle?

Solution:

As we know,
For three line segments to form a triangle, it should satisfy the inequality theorem.
Thus, we have to check whether the um of the two sides is greater than the third side
6 + 12 = 18 and 4 < 18 => True
12 + 4 =16 and 6 < 16 => True
6 + 4 = 10 and 10 < 12 => True
All the conditions of the triangle inequality theorem are satisfied, thus triangle with sides 6cm, 12cm and 4cm needs to be made.

Can a triangle be made with sides 2cm, 3 cm, and 6 cm?

Solution:

As we know,
For three line segments to form a triangle, it should satisfy the inequality theorem.
Here the three given sides are, 2cm, 3 cm, and 6 cm
Thus,
2 + 3 = 5 and 5 < 6 => True
3 + 6 = 9 and 9 > 2 => False
2 + 6 = 8 and 8 > 3 => False
Hence, all the conditions of the triangle inequality theorem are not satisfied.

If the two sides of a triangle are 4 and 9. Find all the possible lengths of the third side.

Solution:

Let the length of the third side be x units.
As we know,
Difference of two sides < unknown side < sum of the two sides
9 – 4 < x < 9 + 4
5 < x < 13
Thus, the third side could have any value between 5 and 13.

Triangle Inequality Theorem Activity

Let us do a simple activity to use the above theorem.

Take three straws of different lengths and colors. Let their colors be red, blue and green in color. Let the red straw measure 4 cm and the blue straw 10 cm, while the length of the green straw is unknown. Arrange the three straws in the form of a triangle, such that the side AB is represented by red straw, side BC by blue straw, and side AC by green straw.

What do you think is the length of the green straw?

To find the possible values of the third side, we can use the given formula,

Difference of two sides < unknown side < sum of the two sides

Let x be the length of the unknown side,

Thus,

10-6 < x < 10 + 6

4 < x < 16

Hence, the length of x will be more than 4 cm and less than 16 cm

Repeat this experiment with different combination of the three straws.

What have you observed?

We observe that the sum of the lengths of any two sides of a triangle is greater than the third side.