# Triangle Inequality – Explanation & Examples

In this article, we will learn what the **triangle inequality theorem** is, how to use the theorem, and lastly, what reverse triangle inequality entails. At this point, most of us are familiar with the fact that a triangle has three sides.

The **three sides of a triangle** are formed when three different line segments join at the vertices of a triangle. In a triangle, **we use the small letters a, b and c to denote a triangle’s sides**.

In most cases, letter **a and b** are used to represent the first **two short sides** of a triangle, whereas letter **c** is used to represent **the longest side**.

## What is Triangle Inequality Theorem?

**As the name suggests, the triangle inequality theorem is a statement that describes the relationship between the three sides of a triangle. According to the triangle inequality theorem, the sum of any two sides of a triangle is greater than or equal to the third side of a triangle. **

*This statement can symbolically be represented as;*

- a + b > c
- a + c > b
- b + c > a

Therefore, a triangle inequality theorem is a **useful tool for checking whether a given set of three dimensions will form a triangle or not**. Simply put, it will not form a triangle if the above 3 triangle inequality conditions are false.

*Let’s take a look at the following examples:*

*Example 1*

Check whether it is possible to form a triangle with the following measures:

4 mm, 7 mm, and 5 mm.

__Solution__

Let a = 4 mm. b = 7 mm and c = 5 mm. Now apply the triangle inequality theorem.

a + b > c

⇒ 4 + 7 > 5

⇒ 11> 5 ……. (true)

a + c > b

⇒ 4 + 5 > 7

⇒ 9 > 7…………. (true)

b + c > a

⇒7 + 5 > 4

⇒12 > 4 ……. (true)

Since all three conditions are true, it is possible to form a triangle with the given measurements.

*Example 2*

Given the measurements; 6 cm, 10 cm, 17 cm. Check if the three measurements can form a triangle.

__Solution__

Let a = 6 cm, b = 10 cm and c = 17 cm

By triangle inequality theorem, we have;

a + b > c

⇒ 6 + 10 > 17

⇒ 16 > 17 ………. (false, 17 is not less than 16)

a + c > b

⇒ 6 + 17 > 10

⇒ 23 > 10…………. (true)

b + c > a

10 + 17 > 6

17 > 6 ………. (true)

Since one of the conditions is false, therefore, the three measurements cannot form a triangle.

*Example 3*

Find the possible values of x for the triangle shown below.

__Solution__

Using the triangle inequality theorem, we get;

⇒ x + 8 > 12

⇒ x > 4

⇒ x + 12 > 8

⇒ x > –4 ……… (invalid, lengths can never be negative numbers)

12 + 8 > x

⇒ x < 20 Combine the valid statements x > 4 and x < 20.

4 < x < 20

Therefore, the possible values of x are; 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19.

*Example 4*

The dimensions of a triangle are given by (x + 2) cm, (2x+7) cm, and (4x+1). Find the possible values of x that are integers.

__Solution__

By the triangle inequality theorem; let a = (x + 2) cm, b = (2x+7) cm and c = (4x+1).

(x + 2) + (2x + 7) > (4x + 1)

3x + 9 > 4x + 1

3x – 4x > 1 – 9

– x > – 8

Divide both sides by – 1 and reverse the direction of the inequality symbol.

x < 8 (x + 2) + (4x +1) > (2x + 7)

5x + 3 > 2x + 7

5x – 2x > 7 – 3

3x > 4

Divide both sides by 3 to get;

x > 4/3

x > 1.3333.

(2x + 7) + (4x + 1) > (x + 2)

6x + 8 > x + 2

6x – x > 2 – 8

5x > – 6

x > – 6/5 …………… (impossible)

Combine the valid inequalities.

1.333 < x <8

Therefore, the possible integer values of x are 2, 3, 4, 5, 6, and 7.

## Reverse Triangle Inequality

**According to reverse triangle inequality, the difference between two side lengths of a triangle is smaller than the third side length**. In other words, any side of a triangle is larger than the subtracts obtained when the remaining two sides of a triangle are subtracted.

Consider triangle *PQR *below;

*The reverse triangle inequality theorem is given by;*

|PQ|>||PR|-|RQ||, |PR|>||PQ|-|RQ|| and |QR|>||PQ|-|PR||

Proof:

- |PQ| + |PR| > |RQ| // Triangle Inequality Theorem
- |PQ| + |PR| -|PR| > |RQ|-|PR| // (i) Subtracting the same quantity from both side maintains the inequality
- |PQ| > |RQ| – |PR| = ||PR|-|RQ|| // (ii), properties of absolute value
- |PQ| + |PR| – |PQ| > |RQ|-|PQ| // (ii) Subtracting the same quantity from both side maintains the inequality
- |PR| > |RQ|-|PQ| = ||PQ|-|RQ|| // (iv), properties of absolute value
- |PR|+|QR| > |PQ| //Triangle Inequality Theorem
- |PR| + |QR| -|PR| > |PQ|-|PR| // (vi) Subtracting the same quantity from both side maintains the inequality
- |QR| > |PQ| – |PR| = ||PQ|-|PR|| // (vii), properties of absolute value