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Let be P the centroid of triangle ABC. If triangle ABP is an equilateral triangle with a side length of 2, then find the perimeter of triangle ABC.

 Mar 16, 2020
 #1
avatar+531 
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Let be P the centroid of triangle ABC. If triangle ABP is an equilateral triangle with a side length of 2, then find the perimeter of triangle ABC.

 

AB = 2

PF = sqrt(3)

CF = PF * 3

AC = BC = sqrt[(AF)² + (CF)²]

Perimeter of triangle ABC = AB + AC + BC    indecision

 Mar 16, 2020
edited by Dragan  Mar 16, 2020
edited by Dragan  Mar 25, 2020
edited by Dragan  Mar 25, 2020
 #2
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Dragan, could you please give your explanation and format your answer as a root? Please and thank you!

 Mar 25, 2020
 #3
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\(4\sqrt7+2\text{ is the answer}\)(I assume this is an AoPS challenge problem and thus you don't need any information. If you do, then remember how 1:2 ratios work with centroids)

 Mar 25, 2020
 #4
avatar+109326 
+1

Using Dragan's  illustration.....

 

AP    =  2

 

And 

 

sin (60°)  = PF /  AP

 

sin(60°)  = PF / 2  ⇒   PF  = √3

 

But CF is a median of  triangle  ABC  and  the PF  is 1/3 of the altitude FC  = 3√3   = √27

 

And  using the Pythagorean Theorem

 

CB  =√ [ FC^2  + FB^2 ]=   √ [27 + 1 ]  =   √28  =   2√7  =   CA

 

So  the perimeter  =  CB + CA  + AB =    2√7  + 2√7 +  2  =    4√7  + 2

 

 

cool cool cool

 Mar 25, 2020

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