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In a triangle with side lengths 5, 6 and $x$, what is the sum of all possible integral values of $x$?

 

 Jun 27, 2018
 #1
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whoops, that diagram was from another question; ignore it, please.

 Jun 27, 2018
 #2
avatar+94209 
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5 , 6  and  x

 

By the triangle inequality, we have two situations

 

We have that

 

5 + 6  >  x

11 > x

So   x  < 11

 

And

 

5  + x  > 6

x  > 1

 

So...we have the following dual inequality

 

1   <  x <  11

 

So....the possible integer values for x are   2,3,4,5,6,7,8,9,10

 

And the sum of these  is  just   (10) (11)  / 2   - 1  =   55 - 1   = 54

 

 

cool cool cool   

 Jun 27, 2018

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