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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.

 

The answer is not 3sqrt3!

 Dec 22, 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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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 23, 2020
 #2
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Just a quick question, how did you form the equations QR = 2 (OQ * sin∠QOP) and h = sqrt[PQ2 - (QR/2)2] = 12*sqrt(3)? 

ianw11  Dec 23, 2020
 #3
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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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

OQ = OR = r = 4√3

 

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

 

QR = 2 (OQ * sin∠QOP) ==>         QR = 3.586301889 

 

Height of ΔPQR      h = sqrt[PQ2 - (QR/2)2= 3.105828542

 

[PQR] = 1/2(h * QR) = 5.569219383 u2

 

jugoslav  Dec 23, 2020
 #5
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Please note ianw11

Answer 1 is not correct, not all of it anyway.

I am pleased that you gave it a go though guest, and some of it is correct.   laugh

Melody  Dec 23, 2020
 #4
avatar+112000 
+3

Thanks Jugoslav, I get exactly the same answer (although I would have left it as an irrational number)

 

Here is my pic - drawn to scale

 

use the formula

 

A = 0.5ab*sinC   to find the area of triangle PQR

 

 Dec 23, 2020
edited by Melody  Dec 23, 2020
 #6
avatar+112000 
+1

Why have you asked this same question twice in two days you little  ##@%##$%#@@#  ?

You have 9 answers on one thread and 5 on the other.

You are totally out to waste everybody's time! 

 

And jugoslav, why have you fully answered on both threads without even commenting?  

 Dec 23, 2020
 #10
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Here I wanted to correct answer #1. I copied and pasted the whole answer and have corrected what was wrong.

I apologize if I've done something wrong, teacher.laugh

Guest Dec 23, 2020
 #11
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I actually do not know who answered that.  That is the problem when people post as guests.

I cannot comment because I do not know which answerer I am talking to.

No do I know which answer/s you are referring to.  

Melody  Dec 24, 2020
edited by Melody  Dec 24, 2020
 #12
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I am only on as a guest because i tried to register for an account at the end it said to check my email for account conformation or something and i did but there was nothing in my email and when i tried to log in it said invalid username or password......

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-Grace

Guest Dec 24, 2020
 #13
avatar+112000 
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Thanks Grace, 

I have let admin know.

In the mean time could you please sign your name 'Grace' at the end of your posts so we know who you are.

 

And don't worry, you haven't done anything wrong.

I was really only upset with the asker.   laugh

Melody  Dec 24, 2020
 #14
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+1

Ok thanks melody! I will sign my name from now on!

-Grace smiley

 Dec 24, 2020
 #15
avatar+112000 
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Thanks Grace  :)

Melody  Dec 24, 2020

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