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1.    In $\triangle ABC$, the acute angles are $\angle B=60^\circ$ and $\angle C=30^\circ$. The centroid of $\triangle ABC$ is $G$. If $AB=1,$ what is $AG?$

 

 

2.   In the diagram, $G$ is the centroid of $\triangle ABC.$ If $[ABC]=72,$ then what is $[GME]?$

 

 

 Nov 24, 2019
 #1
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from my aops textbook

i do not have answer key

 Nov 24, 2019
 #2
avatar+10795 
+1

1.    In $\triangle ABC$, the acute angles are $\angle B=60^\circ$ and $\angle C=30^\circ$. The centroid of $\triangle ABC$ is $G$. If $AB=1,$ what is  

       $AG?$

laugh

.
 Nov 24, 2019
 #3
avatar+2490 
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Because G is the centroid, that is the balancing point, so according to my intuition, the triangle ABC is divided into three congruent triangles (in area) CGA, AGE, and BGC.


Does triangle DFE also have the same centroid as ABC?

 

If so, my intution is telling me that DFE is one-fourth of ABC.

 

But don't trust me, this is intuition.

 Nov 24, 2019
 #4
avatar+212 
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So... triangle DFE is the medial triangle of triangle ABC. Therefore, it has 1/4 of the area of triangle ABC, meaning the area of DFE is 18.

 

I don't know where to go from there.

 Nov 24, 2019
 #5
avatar+2490 
0

DEM is half of DFE.

 

So 18 / 2 = 9.

 

And since G is 2/3 its way on the median.

 

That means GEM is 1/3 of DEM.

 

So.... 3????????????????????????????????????????????????????

CalculatorUser  Nov 24, 2019

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