Triangle inequality

Introduction 

Mathematics seems challenging sometimes, yet it is a fascinating subject. It’s just a matter of understanding it. This topic, triangle inequality, is a very basic chapter of geometry. However, this chapter includes the various basic concepts of geometrical formulas that might help you understand the concept of mathematics better. 

This chapter also contains some important formulae to help understand other geometry chapters. This topic will cover several aspects such as theorems, formulae, and various examples to make it more understandable. 

Concept of the triangle inequality 

Triangle Inequality Theorem states that the shortest distance between two points is always a straight line. This basic concept of geometry claims that the sum of two sides of a triangle is greater than or equal to the third side. Equality holds if three vertices of a triangle become collinear. Symbolically, a+b > c. The Triangle Inequality Theorem has counterparts for metric spaces or some spaces that measure distance. These measures are called Norms which indicate the enclosing of an entity from space in a single or pair or a double vertical line. For a real numerical example, a and b as the absolute value as the norm, obey a specific version of Triangle Inequality written below.

|a| + |b| ≥ |a + b|.

Similarly, a vector space norm obeys a version of triangle inequality for vectors such as x and y. The example is given below. 

||x|| + ||y|| ≥ ||x + y||. With the help of this theorem, the appropriate norms hold for  triangle inequality, complex numbers , integrals or other abstract space.  

Triangle Inequality Theorem

The Triangle Inequality Theorem describes the relationship among the triangle’s three sides.  According to this theorem, the addition of length of any 2 sides of a triangle is greater than or equal to length of the other remaining sides. The below-mentioned information  will state this theorem. 

Let’s suppose a triangle  ABC  has three lengths a, b, and c.  

The theorem states that, a < b + c ; b < a + c  ; c < a + b.

Reverse triangle inequality

The reverse triangle inequality is the 2nd theorem of the triangle inequality. This theorem states that the difference between the lengths of any two sides of a triangle is smaller than the third side of that triangle. Or, in a reverse way, it can be said that one side of any triangle is bigger than the difference between the other two sides.  This means ABC is a triangle, and the sides are AB, AC, and CB, then AB – AC < CB. 

We know the shortest distance between two points is the straight line that connects them. So let us imagine, a straight line of a triangle is being constructed, and that is just a straight line between two points. The other two sides are the other two paths among these two points. However, the two paths go through the third point. Therefore, these paths are not straight, and the other path is longer than the third one. Therefore, the sum of the two sides will be longer from the alternate path than the straight line. 

Example of triangle equality 

To understand the Triangle inequality through practical experience, then just imagine a triangle named ABC. Imagine that you are walking along the sides of the triangle. For example, if you want to go from A to B, then the shortest path will be segment AB.  Now, if you first go to C and then to B, the distance you cover will be AC+ CB, and surely the sum will be greater than AB. 

It can also be proved by drawing the geometric figure. 

• First, let’s draw a segment, AB of Z length.

• Then the second step is to keep the point of your compass to point A and draw an arch of length X

• Keep the tip of the compass on point B, and draw an arc in the same way of length Y. it must intersect the previous arc. 

• Now you get your third point C, which is the intersection point. Now join point B and A with Point C, and get your triangle. 

Now observe carefully that only the arcs will intersect if the other two sides are greater than the distance between the centers of the arc. Therefore, X+Y must be greater than Z. Thus, the theory is appropriate on triangle inequality complex numbers.

Triangle inequality complex numbers 

This is a theorem about distance and approximation for complex real numbers too.  As an example, if we take A = -5 and B = 3, then according to the theory, it will be

|A+ B| = |-5 + 3| = |-2| = 2

|A| + |B|  = |-5 |+ |3| = 5+ 3= 8

therefore, |A+B| < |A| + |B|

Conclusion 

However, It can be concluded that this chapter has the basic concept of geometrical drawing and calculation. This chapter is also significant for understanding the theorems of geometry. This chapter has more segments and facts that need to be learned. The basic chapters are more important to understand your preparation.