In the diagram, $P$ lies on the $y$-axis, $Q$ has coordinates $(4,0)$, and $P Q$ passes through the point $R(2,4)$. What is the area of $\triangle O P Q$ ?
(A) 8
(B) 12
(C) 32
(D) 24
(E) 16

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-039.jpg?height=360&width=423&top_left_y=1918&top_left_x=1293)
The line through $R(2,4)$ and $Q(4,0)$ has slope $\frac{4-0}{2-4}=-2$.

Since it passes through $(4,0)$, this line has equation $y-0=-2(x-4)$ or $y=-2 x+8$.

The line with equation $y=-2 x+8$ has $y$-intercept 8 , and so the coordinates of $P$ are $(0,8)$.

Now, $\triangle O P Q$ is right-angled at $O$ and so its area is $\frac{1}{2}(O Q)(O P)=\frac{1}{2}(4)(8)=16$.

ANSWER: (E)