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June 13th, 2016, 05:46 PM   #1
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Geometry Triangle Question

1. In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there?



2. 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?

Please Help. I tried using triangle inequality, but I got nowhere...

Last edited by skipjack; June 13th, 2016 at 11:34 PM.
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June 13th, 2016, 06:29 PM   #2
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1). $\angle B > \angle C \implies 3n+1 > 4n-9 \implies n < 10$

Triangle inequality ... $7n-8 > 3n+4 \implies n > 3$

conclusion?

2). If 8 and 15 are the two shorter sides and x is the longest side, then $8^2+15^2 > x^2 \implies x < 17$

If 8 and x are the two shorter sides and 15 is the longest side, then $8^2+x^2 > 15^2 \implies x > \sqrt{161} \approx 12.7$

conclusion?
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June 13th, 2016, 06:42 PM   #3
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But #2 says it is a positive integer
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June 13th, 2016, 11:26 PM   #4
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And it shows possible values.
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June 14th, 2016, 04:42 AM   #5
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Quote:
Originally Posted by Oran2009 View Post
But #2 says it is a positive integer
How many positive integers are within the interval $\sqrt{161} < x < 17$ ?
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