Triangle Inequality Theorem Proof
In this section, we will learn the proof of the triangle inequality theorem. To prove the theorem, assume there is a triangle ABC in which side AB is produced to D and CD is joined.
Triangle Inequality Theorem Proof:
Notice that the side BA of Δ ABC has been produced to a point D such that AD = AC. Now, since ∠BCD > ∠BDC.
By the properties mentioned above, we can conclude that BD > BC.
We know that, BD = BA + AD
So, BA + AD > BC
= BA + AC > BC
So, this proved sum of two sides triangle is always greater than the other side.
Let’s see an example based on Triangle Inequality Theorem to understand its concept more clearly.
Example: D is a point on side BC of triangle ABC such that AD = DC. Show that AB > BD.
Solution:
In triangle DAC,
AD = AC,
∠ADC = ∠ACD (Angles opposite to equal sides)
∠ ADC is an exterior angle for ΔABD.
∠ ADC > ∠ ABD
⇒ ∠ ACB > ∠ ABC
⇒ AB > AC (Side opposite to larger angle in Δ ABC)
⇒ AB > AD (AD = AC)
Check: Types of Triangles
Triangle Inequality Theorem, Proof & Applications
Triangle Inequality Theorem is the relation between the sides and angles of triangles which helps us understand the properties and solutions related to triangles. Triangles are the most fundamental geometric shape as we can’t make any closed shape with two or one side. Triangles consist of three sides, three angles, and three vertices.
The construction possibility of a triangle based on its side is given by the theorem named “Triangle Inequality Theorem.” The Triangle Inequality Theorem states the inequality relation between the triangle’s three sides. In this article, we will explore the Triangle Inequality Theorem and some of its applications as well as the other various inequalities related to the sides and angles of triangles.
In this article, we’ll delve into the concept of triangle inequality, the triangle inequality theorem, its significance, and its practical applications.
Table of Content
- What is Triangle Inequality Theorem?
- Triangle Inequality Theorem Formula
- Triangle Inequality Theorem Proof
- Triangle Inequality Theorem – Applications & Uses
- How to Identify Triangles
- How to Find Range of Possible Values of Sides of Triangle
- Various Inequalities in Triangle
- Sample Problems on Triangle Inequality Theorem
- Practice Problem on Triangle Inequality Theorem