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# hi :) some math

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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.

Dec 28, 2020

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A circle is centered at O  and has an area of 48pi. 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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OQ = OR = r = 4√3

∠QOR = 30º         ∠QOP = 15º           ∠QPR = 60º

QR = 2 (OQ * sin∠QOP) ==>         QR = 6*sqrt(3)

Height of ΔPQR      h = sqrt[PQ2 - (QR/2)2] = 12*sqrt(3)

[PQR] = h * QR / 2 = 8*sqrt(3)

Dec 28, 2020
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I'm sorry but that answer is incorrect

Guest Dec 28, 2020
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Using this diagram, you should be able to find the area of the equilateral triangle QRP.

Area of a triangle QRP = $$\sqrt[3]3$$

Credit to Dragan; https://web2.0calc.com/members/dragan/

Dec 28, 2020
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Hi, cryptoaops!

That diagram is ok, but my answer "Area of a triangle QRP = 3√3" is NOT correct!!!

The correct answer is   [QRP] = 5.569219381 square units!!!