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I came across this question while I was taking one of the pratice Mu Alpha Theta tests for my school and I wasn't sure how to solve it. It reads: In $\Delta USA $, $\angle S$ is bisected by $\overrightarrow {SY}$, with $Y$ on side $\overline {UA}$. If all sides have integer values, $ \overline{US}=18 $ and $ \overline{UY} = 12 $; find the smallest possible perimeter of $\Delta USA$.

The answer key with the solutions skipped to the part with the triangle inequalities, so that is why I am not sure what to do. I think there is some ratio when it comes to splitting a triangle with an angle bisector but I am not sure what it is. My question is how would I solve this question, and for future reference what is that ratio for an angle bisector in a triangle.

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up vote 2 down vote accepted

Hint: The relevant theorem about angle bisectors (adapted to your labels) is $$\frac{AY}{YU}=\frac{SA}{SU}.$$

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Oh then you could, find the lengths, and then using The Triangle Inequality find what the shortest possible lengths could be. Thank You! – Andrew Delgadillo Dec 1 '12 at 20:25

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