Triangle Inequality: Activity-Triangle Inequality Theorem and Triangle Inequality Theorem

Get unlimited access to the best preparation resource for JEE/Mains Mathematics: fully solved questions with step-by-step explanation- practice your way to success.

Download PDF of This Page (Size: 183K)

A triangle is a three-sided polygon. It has three sides and three angles. The three sides and three angles share an important relationship. Term “inequality” represents the meaning “not equal”.

What is Triangle Inequality?

“Triangle inequality” is meant for any triangles. Let us take a, b, and c are the lengths of the three sides of a triangle, in which no side is being greater than the side c, then the triangle inequality states that,

This states that the sum of any two sides of a triangle is greater than or equal to the third side of a triangle.

Activity - Triangle Inequality Theorem

Activity 1: On a paper mark two points Y and Z and join them to form a straight line. Mark another point X outside the line lying on the same plane of the paper. Join XY as shown.

Now mark another point X’ on the line segment XY, join X’Z. Similarly mark X’’ and join X’’Z with dotted lines as shown.

Activity: Triangle Inequality Theorem

Activity: Triangle Inequality Theorem

Loading Image

From the above figure we can easily deduce that if we keep on decreasing the length of side XY such that XY> X’Y> X’’Y> X’’’Y the angle opposite to side XY also decreases i.e. ∠XZY >∠X’ZY >∠X’’ZY >∠X’’’ZY. Thus, from the above activity we can infer that if we keep on increasing one side of a triangle then the angle opposite to it increases.

Now let us try out another activity.

Activity: Draw 3 scalene triangles on a sheet of paper as shown.

Activity: Triangle Inequality Theorem

Activity: Triangle Inequality Theorem

Loading Image

Let us consider fig. (i). In is the longest side and is the shortest. We observe that is the largest in measure and is the smallest. Similarly, in is the largest side and is the smallest and is the largest in measure and is the smallest. In the last figure also the same kind of pattern is followed i.e. side is largest and so is the opposite to it.

Triangle Inequality Theorem

Let us consider the triangle. The following are the triangle inequality theorems.

Triangle Inequality Theorem

Triangle Inequality Theorem

Loading Image

Theorem 1: In a triangle, the side opposite to the largest side is greatest in measure.

The converse of the above theorem is also true according to which in a triangle the side opposite to a greater angle is the longest side of the triangle.

In the above fig., since is the longest side, the largest angle in the triangle is

Another theorem which follows can be stated as:

Theorem 2: The sum of lengths of any two sides of a triangle is greater than the length of its third side.

According to triangle inequality,

Example: In and , which side of the triangle is the longest?

Solution:

  • Here given,

  • In

  • For the finding of other triangle, using angle sum property of triangle.

Put the value of in above equation.

Sum of the angle P and angle Q.

  • So, is the largest angle.

  • According to Theorem 1, In a triangle, the side opposite to the largest side is greatest in measure.

  • Here largest angle is , opposite side of the triangle is the largest side.

  • is the longest side.