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# Triangle Inequality

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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?

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?

Apr 25, 2022

#1
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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   Apr 25, 2022
#2
+1

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   Apr 25, 2022