Triangle Inequality

A triangle has integer length sides.  If two sides of the triangle are 18 and 44 , how many possible lengths are there for the third side?

18  + x >  44           18 + 44 > x

 x > 44 - 18               62  > x

x >  26

 So    

26 < x <  62  ......number of possible integer lengths =  61  - 27  + 1   =  35

A triangle has integer length sides.  If two sides of the triangle are 18 and 44 , and the triangle is obtuse, how many possible lengths are there for the third side?

If obtuse

18^2  + x^2  <  44^2                                           44^2  + 18^2  < x^2

x^2  <  44^2  -18^2                                             2260 < x^2

x^2 < 1612                                                          ceiling sqrt (2260)  <  x  =  48

x < floor  sqrt (1612)  = 40   

So     48  -  40  + 1   =   9  possible integer lengths