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Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side, if it is a positive integer?

 Apr 19, 2019
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We have the following inequalities

 

8 + 15 >  x        where x is the unknown side

23 > x

x < 23         and

 

8 + x >  15

x > 7

 

However....since the triangle is acute....we know that  an 8 - 15 - 17  triangle is a right triangle

 

So....the remaining side, x, must be <  17  [  any integer > 17  but < 23  will produce an obtuse triangle ]

 

So....the possible side lengths are integers  between 7  and  17

 

However....we must even restrict this interval  to  integers  greater than 12 and < 17 

 

The reason for this is that integer side lengths for x  from 8 to 12 inclusive also produce obtuse triangles 

 

So....4 possible acute triangles

 

 

cool cool cool

 Apr 20, 2019

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