In the diagram, points $P, Q, R$, and $S$ are arranged in order on a line segment. If $P Q=1, Q R=2 P Q$ and $R S=3 Q R$, then the length of $P S$ is
(A) 7
(B) 6
(C) 9
(D) 8
(E) 10

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-080.jpg?height=92&width=430&top_left_y=884&top_left_x=1292)
Solution 1

Since $P Q=1$ and $Q R=2 P Q$, then $Q R=2$.

Since $Q R=2$ and $R S=3 Q R$, then $R S=3(2)=6$.

Therefore, $P S=P Q+Q R+R S=1+2+6=9$.

## Solution 2

From the given information,

$$
P S=P Q+Q R+R S=P Q+Q R+3 Q R=P Q+4 Q R=P Q+4(2 P Q)=9 P Q
$$

Thus, $P S=9(1)=9$.

ANswer: (C)