, 2014

Since $\triangle PQR$ has $PQ=QR=RP$, then $\triangle PQR$ is equilateral and all of its angles equal $60^\circ$. Since $ST$ is parallel to $QR$, $SV$ is parallel to $PR$, and $TU$ is parallel to $PQ$, then all of the angles in $\triangle PST$, $\triangle SQV$ and $\triangle TUR$ equal $60^\circ$. In other words, each of these triangles is also equilateral. Let $SQ=x$. Since $\triangle SQV$ is equilateral, then $QV=VS=SQ=x$. Since $PQ=30$, then $PS=30-x$. Since $\triangle PST$ is equilateral, then $ST=TP=PS=30-x$. Since $PR=30$, then $TR=30-(30-x)=x$. Since $\triangle TUR$ is equilateral, then $TU=UR=TR=x$. [[IMAGE0]] Since $VS+ST+TU=35$, then $x+(30-x)+x=35$ or $30+x=35$ and so $x=5$. Therefore, $VU=QR-QV-UR=30-x-x=30-5-5=20$.