In the diagram, $O$ is the centre of a circle with radius 87 , and $P$ and $M$ lie on the circle. $N$ is positioned inside the circle so that $P N$ passes through $O$ and is perpendicular to $M N$. If $M N=63$, what is the area of $\triangle P M N$ ?
(A) 3370.5
(B) 3496.5
(C) 4725.0
(D) 4630.5
(E) 4126.5

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-004.jpg?height=352&width=372&top_left_y=992&top_left_x=1321)
Since $O M$ is a radius of the circle, then $O M=87$.

$\triangle M N O$ is a right-angled triangle, and so by the Pythagorean Theorem, we get $O M^{2}=M N^{2}+N O^{2}$ or $87^{2}=63^{2}+N O^{2}$, and so $N O^{2}=87^{2}-63^{2}=3600$.

Since $N O>0$, then $N O=\sqrt{3600}=60$.

Since $O P$ is also a radius, then $O P=87$, and so $N P=N O+O P=60+87=147$.

The area of $\triangle P M N$ is equal to $\frac{1}{2} \times N P \times M N=\frac{1}{2} \times 147 \times 63=4630.5$.

ANSWER: (D)