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In the triangle shown, for \(\angle A\) to be the largest angle of the triangle, it must be that m < x < n. What is the least possible value of   n - m,expressed as a common fraction?

 Jun 22, 2019
 #1
avatar+7725 
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What is m and n? The problem is still not complete!

Please mind what you are posting, do not just copy from some website and throw it here.

At least check if there are any mistakes in the problem before posting... *facepalm*

 Jun 22, 2019
 #2
avatar+8810 
+3

In order for ∠A to be the largest angle, It must be that the side formed by ∠A is the largest side. So...

 

3x  <  x + 9  
2x  <  9

 

 

x  <  \(\frac92\)  

 

But by the triangle inequality theorem,

 

x + 9  <  ( 3x ) + ( x + 4 )  
x + 9  <  3x + x + 4

 

 

9  <  3x + 4  
5  <  3x

 

 

\(\frac53\)  <  x  

 

So in order for ∠A to be the largest angle, it must be that   \(\frac53
 

The least possible value of  n - m  is  \(\frac92-\frac53\ =\ \frac{27}{6}-\frac{10}{6}\ =\ \boxed{\frac{17}{6}}\)

 

( Here's an older answer for this question: https://web2.0calc.com/questions/geometry-question. . .#r5 )

 Jun 22, 2019

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