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Math A

Triangle Inequalities

 

In mathematics our focus is oftentimes the same as our country's founders'......equality.


But, in mathematics, there are also important relationships which deal with unequal quantities.  Let's examine two of these relationships.


 
  Theorem

The first relationship involves the lengths of the sides of a triangle.

The sum of the lengths of any two sides of a triangle must be greater than the
third side.
 

 

Example
Suppose we know the lengths of two sides of a triangle, and we want to find the possible lengths of the third side.
According to our theorem, the following 3 statements must be true:

   5+x > 9
   So, x > 4    

    5+9 > x
    So, 14 > x     

x+9 > 5
So, x > -4   
(no real information gained here since the lengths of the sides must be positive.)
Putting these statements together we get that x must be greater than 4, but less than 14.  So any number in the range 4 < x < 14 can represent the length of the missing side of our triangle.

 

 

The second relationship involves the lengths of the sides of a triangle in relation to the triangle's angles.
  Theorem
 

In a triangle,
the longest side
 is across from the largest angle.

 

Example
Suppose we want to know which side of this triangle is the longest.
Before we can utilize our theorem, we need to know the size of <B.  We know that the 3 angles of the triangle add up to 180º. 80 + 40 + x = 180
     120 + x = 180
             x = 60
 

We have now found that <B measures 60º.  According to our theorem, the longest side will be across from the largest angle. 

Now that we know the measures of all 3 angles, we can tell that <A is the largest.  This means the side across from <A, side CB, is the longest.

 

While there are other inequality relationships in a triangle,
 these two relationships are the ones most commonly used.
Be sure that you learn these two relationships and you'll be set !

 


   Murray