Triangle inequality

Introduction

The triangle inequality theorem tells us about the correlation between the triangle’s three sides. Let us suppose a triangle with the length of the sides given as a, b and c, then as stated in the theorem, the sum of two sides will be greater than or could be equal to the third side of the triangle, which implies that:

a<b+c

c<a+b

b<a+c

In general terms, this theorem states that the distance between distinct two points is always a straight line. The important point is that inequality is applicable only when the triangle is non-degenerate. Triangle Inequality Theorem also states that the side opposite to the largest side of the triangle is the longest. The converse also holds. 

Proof of Triangle Inequality Theorem

We take a triangle  △ABC. Extend the side AC to a point D such that AD = AB

Now we have |CD|=|AC|+|AD|        

And |CD|= |AC|+|AB|    (since AD=AB which means △ADB is an isosceles triangle)

∠DBA<∠DBC       (because ∠DBC = ∠DBA+∠ABD)    —(1)

This implies ∠ADB<∠DBC     (since it is given that ∆ADB is an isosceles triangle and ∠ADB = ∠DBA)                                                                               —(2)

Now, |BC|<|CD|    (From the rule that the side opposite the larger angle is greater)

Therefore from (1) and (2) we come to the conclusion that 

|BC|<|AC| + |AB|   

Triangle Inequality in Vector Form

Triangle Inequality in vector form can be formulated as:

Let p and q be vectors in Rn . The the triangle inequality is given by

|p| – |q| ≤  |p + q| ≤ |p| + |q|

Special case of Right Triangles

The triangle inequality is a corollary of the Pythagorean theorem for right triangles and the law of cosines for generic triangles in Euclidean geometry, though it can be proved without these theorems. Intuitively, the inequality can be seen in R2 or R3. Only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, does equality exist in the Euclidean situation. The shortest distance between two places in Euclidean geometry is hence a straight line.

Solved Examples 

Example1

We have been given three line segments, 4cm, 2cm and 8cm. Check whether they can be used to form a triangle.

Solution

To form a triangle it should satisfy the triangle inequality property.

We check all the possibilities.

4 + 8 > 2 ⇒ 12 > 2 ⇒ True

8 + 2 > 4 ⇒ 10 > 4 ⇒ True

4 + 2 > 8 ⇒ 6 > 8 ⇒ False

One of the possibilities does not satisfy the Triangle inequality theorem, therefore we cannot form a triangle with the given line segments.

Example2

Two sides of a non-degenerate triangle are given as 9cm and 15cm. Find the possible lengths of the third side of the triangle.

Solution

Let the length of the third side of the triangle be x (>0)

The lengths of the triangle must satisfy the Triangle Inequality property. So we use the formula 

Difference of two sides

15-9<x<15+9

6<x<24

The length of the third will be between 6cm and 24cm.

Conclusion

In this article, we learned about the Triangle Inequality property, which states that the sum of any two different sides of a triangle is larger than the third side. Thus, it means that the shortest distance between any two distinct points is always a straight line. Civil engineers employ the triangle inequality theorem in the real world because their work involves surveying, transportation, and urban planning. They can use the triangle inequality theorem to calculate