# Math Facts

### Why triangle inequality is so damn important!

Triangle inequality is an important triangle property that is *very* frequently tested on the GRE. Remember this:

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

The above figure shows $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \triangleABC$ $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \triangle ABC$with sides $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a$, $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} b$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} c$ as shown. Based on what we just learnt, all of the following inequalities are true:

$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a+b>c$

$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a+c>b$

$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} b+c>a$

Example:

It two sides of a triangle are 5 cm and 6 cm respectively, which of the following can NOT be a value of the third side?

A) 2 cm

B) 5 cm

C) 6 cm

D) 10 cm

E) 11 cm

Solution:

The third side has to be less than the sum, and more than the difference of 5 and 6. Therefore the third side must be less than 11 and more than 1.

Thus 11 CM cannot be the length of the third side.