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 June 13th, 2016, 06:46 PM #1 Newbie   Joined: Jun 2016 From: Austin Posts: 2 Thanks: 0 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 14th, 2016 at 12:34 AM.
 June 13th, 2016, 07:29 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,817 Thanks: 1460 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?
 June 13th, 2016, 07:42 PM #3 Newbie   Joined: Jun 2016 From: Austin Posts: 2 Thanks: 0 But #2 says it is a positive integer
 June 14th, 2016, 12:26 AM #4 Newbie   Joined: Jun 2016 From: Poland Posts: 14 Thanks: 2 And it shows possible values.
June 14th, 2016, 05:42 AM   #5
Math Team

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Quote:
 Originally Posted by Oran2009 But #2 says it is a positive integer
How many positive integers are within the interval $\sqrt{161} < x < 17$ ?

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