Triangle Inequality Theorem

A polygon bounded by three line segments is known as the Triangle. It is the smallest possible polygon. A triangle has three sides, three vertices, and three interior angles.

Triangle Inequality Theorem

The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side. In other words, this theorem specifies that the shortest distance between two distinct points is always a straight line.

Consider a ∆ABC as shown below, with a, b and c as the side lengths.

Triangle Inequality

Triangle ABC

The triangle inequality theorem states that:

a < b + c,

b < a + c,

c < a + b

In any triangle, the shortest distance from any vertex to the opposite side is the Perpendicular. In fig. below, XP is the shortest line segment from vertex X to side YZ.

Triangle Inequality

Triangle XYZ

Proof for Triangle Inequality Theorem –

Triangle Inequality

Triangle ABC

To Prove-

|BC|< |AB| + |AC|

Construction: Consider a ∆ABC. Extend the side AC to a point D such that AD = AB as shown in the fig. below.

Triangle Inequality

Proof of triangle inequality theorem

S.No Statement Reason
1. |CD|= |AC| + |AD| From figure 3
2. |CD|= |AC| + |AB| AB = AD, ∆ADB is an isosceles triangle
3. ∠DBA <∠DBC Since ∠DBC = ∠DBA+∠ABD
4. ∠ADB<∠DBC ∆ADB is an isosceles triangle and ∠ADB = ∠DBA
5. |BC|<|CD| Side opposite to larger angle is larger in length
6. |BC|<|AC| + |AB| From statements 3 and 4

Thus, we can conclude that the sum of two sides of a triangle is greater than the third side.

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Practise This Question

Which of the fractions lie between 38 and 78 on the number line?