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In a triangle $\bigtriangleup ABC$ is $\widehat A=30^{\circ}$, $|AB|=10$ and $|BC|\in\{3,5,7,9,11\}$.

How many non-congruent trangles $\bigtriangleup ABC$ exist?

The possible answers are $3,4,5,6$ and $7$.

Is there a quick solution that doesn't require much writing?

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    By $n$ot much writing i mean not $s$olvi$n$g equatio$n$s and writing out numbers above 100.2012-10-05

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Hint: Use the Sine Law: $\frac{\sin C}{10}=\frac{\sin A}{k},$ where $k$ is one of our numbers $3$, $5$, $7$, $9$, $11$. Since $\sin A=1/2$, one of our $k$ is problematical. Another yields a triangle we all know and love. For the others, we are dealing with possibly the "ambiguous" case. A couple of sketches will give the answer, or knowledge about when we really are in the ambiguous case.

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    I am on the other side of the fence, involved in making up questions for a couple of contests. Don't like multiple choice, too many times they are setting traps.2012-10-06