Triangle Inequality Theorem – Applications & Uses
There are many applications in the geometry of the Triangle Inequality Theorem, some of those applications are as follows:
- To Identify the Triangles
- To Find the Range of Possible Values of the Sides of Triangles
How to Identify Triangles
To Identify the possibility of the construction of any given triangle with three sides, we can use the Triangle Inequality Theorem. If the given three sides satisfy the theorem, then the construction of this triangle is possible.
For example, consider the sides of the triangle as 4 units, 5 units, and 7 units.
As 4 + 5 > 7, 5 + 7 > 4, and 7 + 4 > 5. Thus, the triangle with sides 4 unit, 5 units, and 7 units is possible to construct.
Now take another example for not possible construction, consider the sides of the triangle to be 3 units, 4 units, and 9 units.
As 3 + 4 [Tex]\ngtr [/Tex] 9, therefore triangle with sides 3 units, 4 units and 9 units is not possible.
How to Find Range of Possible Values of Sides of Triangle
To find the range of possible values of the sides of a triangle when two sides are given as units and b units, you can follow these steps:
Step 1: Let’s assume the third side be x units.
Step 2: Now, using Triangle Inequality Theorem we know,
a + b > x, a + x > b, and b + x > a
Step 3: Using all three conditions we can find the range for the third side.
Let’s consider an example for better understanding.
Example: Find the range for the third side of the triangle if the first two sides are 4 units and 7 units.
Solution:
Let’s assume the third side be x units.
Using Triangle Inequality Theorem, we get
4 + 7 = 11 > x, 4 + x > 7, and 7 + x > 4
Simplifying the above inequalities, we get
11 > x and x > 3.
Thus, possible range for the third sides is 3 < x < 11.
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