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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
 #1
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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
 #2
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I'm sorry but that answer is incorrect

Guest Dec 28, 2020
 #3
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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
 #4
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Hi, cryptoaops!

 

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

Link:  https://web2.0calc.com/questions/geometry_49612

 

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

Link:   https://web2.0calc.com/questions/quick-hard-question-help-asap

 

(I don't know how to convert that decimal into radical.)

Dragan  Dec 30, 2020

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