Asked by Knights

Two sides of a triangle are 11 and 17. How many possible lengths are there for the third side, if it is a positive integer?

I tried listing them all but I get confused...Please help thank you!
Answered by Reiny
The sum of any two sides must be greater than the third side for a triangle to exist
let the third side be x
x+11>17 AND x+17>11 AND 11+17> x
x > 6 AND x>-6 AND x < 28

so 6 < x < 28

So how many positive integers can you count between 6 and 28 exclusive ?
Answered by Art od Problem Solving
PLEASE DO NOT TRY TO CHEAT AND USE THE INTERNET FOR YOUR ANSWERS TO YOUR HOMEWORK THANKS!
Let the third side be $n$. Then by the triangle inequality,
\begin{align*} n + 11 &> 17, \\ n + 17 &> 11, \\ 11 + 17 &> n, \end{align*}
which gives us the inequalities $n > 6$, $n > -6$, and $n < 28$. Therefore, the possible values of $n$ are 7, 8, $\dots$, 27, for a total of $27 - 7 + 1 = \boxed{21}$ possible values.
Answered by AoPS Admin
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Answered by AoPS Admin
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Answered by JONNY
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