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G is the centroid of triangle ABC and D and E are the midpoints of line AB and line AC, respectively. line AG and line DE intersect at M. Find GM/GA.

beastrohit  Nov 28, 2018
 #1
avatar+92641 
+1

 

AE and AD divide sides AB and AB proprtionally

Therefore....DE is parallel to BC

And angle  FAC = angle MAE

And since AF is a transversal cutting parallel lines DE and BC, then angle AME = angle AFC

So.....byAA concruency, triangle AME is simiar to triangle AFC

But since AE = 1/2 AC....then AM = (1/2) AF

And since G is a centroid, then AG = (2/3)AF

Then GM = AG  - AM  =  (2/3)AF - (1/2)AF = (1/6)AF

 

Therefore   GM : GA  =    (1/6)AF / (2/3)AF =   (1/6) / (4/6) =  1 : 4   

 

 

cool cool cool  

CPhill  Nov 28, 2018

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