In the diagram, $\triangle P Q R$ has $P Q=Q R=R P=30$. Points $S$ and $T$ are on $P Q$ and $P R$, respectively, so that $S T$ is parallel to $Q R$. Points $V$ and $U$ are on $Q R$ so that $T U$ is parallel to $P Q$ and $S V$ is parallel to $P R$. If $V S+S T+T U=35$, the length of $V U$ is
(A) 21
(B) 15
(C) 18
(D) 20
(E) 25

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-058.jpg?height=455&width=486&top_left_y=2036&top_left_x=1226)

## Part C: Each correct answer is worth 8.
Since $\triangle P Q R$ has $P Q=Q R=R P$, then $\triangle P Q R$ is equilateral and all of its angles equal $60^{\circ}$. Since $S T$ is parallel to $Q R, S V$ is parallel to $P R$, and $T U$ is parallel to $P Q$, then all of the angles in $\triangle P S T, \triangle S Q V$ and $\triangle T U R$ equal $60^{\circ}$. In other words, each of these triangles is also equilateral.

Let $S Q=x$.

Since $\triangle S Q V$ is equilateral, then $Q V=V S=S Q=x$.

Since $P Q=30$, then $P S=30-x$.

Since $\triangle P S T$ is equilateral, then $S T=T P=P S=30-x$.

Since $P R=30$, then $T R=30-(30-x)=x$.

Since $\triangle T U R$ is equilateral, then $T U=U R=T R=x$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-115.jpg?height=463&width=509&top_left_y=1384&top_left_x=862)

Since $V S+S T+T U=35$, then $x+(30-x)+x=35$ or $30+x=35$ and so $x=5$.

Therefore, $V U=Q R-Q V-U R=30-x-x=30-5-5=20$.

ANSWER: (D)