In $\triangle P Q R$, a line segment is drawn from $P$ to point $S$ on side $Q R$. If $\triangle P Q S$ and $\triangle P R S$ have the same area, which of the following statements must be true?
(A) $P Q=P R$
(B) $P S=P Q$
(C) $Q R=P S$
(D) $Q S=S R$
(E) $P Q=Q R$

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-057.jpg?height=342&width=415&top_left_y=688&top_left_x=1297)

## Part C: Each correct answer is worth 8.
Position point $T$ on $Q R$ such that $P T$ is perpendicular to $Q R$.

Line segment $P T$ is the height of $\triangle P Q S$ with base $Q S$, and is also the height of $\triangle P R S$ with base $S R$.

Since $\triangle P Q S$ and $\triangle P R S$ have equal heights and equal areas, their bases must be equal. Thus, $Q S=S R$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-268.jpg?height=301&width=366&top_left_y=2278&top_left_x=1541)

ANSWER: (D)