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

 Nov 17, 2019
 #1
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Draw a line through M, which intersect AE at T.  Then GM/GA = ET/EA = 1/3.

 Nov 17, 2019
 #2
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Draw FH  through G   such that  FH  is parallel to   DE.....and let F lie on AB and H lie on AC

 

Then triangles DEA , FHA   and  BCA  are similar

 

And since G is a median  then AG  is  2/3  of the distance  from A to BC

 

So...AF  is 2/3  of AB

And AH  is 2/3 of AC

 

So   DF  = (2/3)AB - (1/2)AB = (1/6)AB

 

So  DF /  AF   =  [ (1/6)AB]  / [(2/3)AB] =   (1/6) / (2/3)  =   1/4

 

And  because triangles  DEA  and FHA are similar....then  DF /AF  =   GM / GA  =  1/4

 

 

cool cool cool

 Nov 17, 2019
edited by CPhill  Nov 17, 2019
edited by CPhill  Nov 17, 2019

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