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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. (3sqrt3 is not the answer)

 Dec 21, 2020
 #1
avatar+428 
-2

OQ = OR = 4√3

 

Circumradius of a triangle = $\frac{ a b c}{\sqrt{(a + b - c) (a - b + c) (-a + b + c) (a + b + c)}}$

 

Inputting 4√3, we have $4\sqrt{3} = \frac{48c}{\sqrt{(8\sqrt{3} - c) (c) (c) (8\sqrt{3} + c)}}$

 

Which simplifies to $4\sqrt{3} = \frac{48c}{\sqrt{(8\sqrt{3} - c)(8\sqrt{3}+c)(c^2)}}$

 

By difference of squares, $4 \sqrt{3} = \frac{48c}{\sqrt{192-c^2)(c^2}} = \frac{48c}{c \sqrt{192-c^2}} = \frac{48}{\sqrt{192-c^2}} = \frac{48 \sqrt{192-c^2}}{192-c^2}.$

 

Cross multiplying, $768 \sqrt{3} - 4c^2 \sqrt{3} = 48 \sqrt{192-c^2}$ and $192 \sqrt{3} - c^2 \sqrt{3} = 12 \sqrt{192-c^2}$ and squaring, $(192-c^2)(192-c^2)(3) = 144(192-c^2),$ so $192-c^2 = 48,$ and consequently, $c=12.$ Thus, 12^2 / 4 * √3 = 144/4 * √3 = 36√3.

 

$\square$

 

Seems like the person who answered forgot to square the 12 :)

 Dec 21, 2020
 #4
avatar+27 
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Hmm... your answer is wrong Pangolin14... not really sure what went wrong though...

ianw11  Dec 22, 2020
 #2
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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] = 3.105828541

 

[PQR] = h * QR / 2 = 5.569219382 u2

 

 Dec 22, 2020
 #3
avatar+428 
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There seem to be two conflicting answers. If I did make a mistake, can you explain what I did wrong?

Pangolin14  Dec 22, 2020
 #5
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Can you express your answer as a sqrt please? 

ianw11  Dec 22, 2020
 #6
avatar+428 
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It is expressed as a square root.

Pangolin14  Dec 22, 2020
 #7
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Sorry, I was addressing jugoslav... his answer may be correct, but his answer is expressed as a decimal. Also jugoslav, how did you form the equation QR = 2 (OQ * sin∠QOP? 

ianw11  Dec 22, 2020
edited by ianw11  Dec 22, 2020
 #8
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0

The closest radical is √31cool I don't know how to find the exact radical.

 

OQ = 4√3         ∠QOP = 15º

 

QR = 2*(OQ * sin∠QOP)   ==> QR = 2*(4√3 * sin15º)       

Guest Dec 23, 2020

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