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 $PQR.$

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virtuematane virtuematane · Answer 1 · 2021-06-18T08:35:59+02:00

Answer:

Area of Triangle QRP = 3[tex]\sqrt{3}[/tex]

Step-by-step explanation:

According to Question , We have a circle With Centre 'O' & Area 48[tex]\pi[/tex] .

Area Of Circle = 48[tex]\pi[/tex]

[tex]\pi[/tex][tex]r^{2}[/tex] = 48[tex]\pi[/tex]

r = [tex]\sqrt{48}[/tex]

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([tex]r_{2}[/tex]) = [tex]\frac{\sqrt{48}}{2}[/tex]

Now We Have To Find Area Of Equilateral Triangle .

A = [tex]\frac{\sqrt{3}}{4} r_{2} ^{2}[/tex]

A= [tex]\frac{\sqrt{3} }{4} * \frac{\sqrt{48} }{2}*\frac{\sqrt{48} }{2}[/tex]

The Area Of PQR is = 3[tex]\sqrt{3}[/tex]

For Diagram , Please Find In Attachment