Question

A circle is centered at $O$ and has an area of $48 \pi.$ Let $Q$ and $R$ be points on the circle, and let $P$ be the circumcenter of triangle $QRO.$ If $P$ is contained in triangle $QRO,$ and triangle $PQR$ is equilateral, then find the area of triangle $PQR.$

Answer 1

Answer:

Area of Triangle QRP = 3$\sqrt{3}$

Step-by-step explanation:

According to Question , We have a circle With Centre 'O' & Area 48$\pi$ .

Area Of Circle = 48$\pi$

$\pi$$r^{2}$ = 48$\pi$

r = $\sqrt{48}$

Now We Have Two Points Given On Circle Q & R , P Is Circumcentre Of Triangle QRO .

Thus A Circle Can Also Be Formed with Centre P . ( See attachment For Diagram )

Now The Diameter of Circle With Centre P = Radius Of Circle with Centre O

so Radius Of Circle With Centre P($r_{2}$) = $\frac{\sqrt{48}}{2}$

Now We Have To Find Area Of Equilateral Triangle .

A = $\frac{\sqrt{3}}{4} r_{2} ^{2}$

A= $\frac{\sqrt{3} }{4} * \frac{\sqrt{48} }{2}*\frac{\sqrt{48} }{2}$

The Area Of PQR is = 3$\sqrt{3}$

For Diagram , Please Find In Attachment