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If G is the centroid of the triangle ABC and O is any other point in the plane then prove that (OA) 2 + (OB) 2 + (OC) 2 =(GA) 2 + (GB) 2 + (GC) 2 + 3(GO) 2 If G is the centroid of the triangle ABC and O is any other point in the plane then prove that (OA)2 + (OB)2 + (OC)2=(GA)2 + (GB)2 + (GC)2 + 3(GO)2
If G is the centroid of the triangle ABC and O is any other point in the plane then prove that (OA)2 + (OB)2 + (OC)2=(GA)2 + (GB)2 + (GC)2 + 3(GO)2
Dear Sohan You can do it by coordinates and vectors both methods vector method is simple and involves less variables assume o as origin Position vectors of A,B,C as a,b,c respectively OA=a, OB=b,OC=c G=a+b+c/3 GA=a-(a+b+c)/3=2a-b-c/3 similarly you can write for GB,GC,GO GA2=(4a2+b2+c2-4a.b+b.c-4a.c)/9 similarly write for others and just calculate the expression similarly using coordinate you can take o as (0,0)and A,B,C as (x1,y1),(x2,y2)and (x3,y3) G will be ((x1+x2+x3)/3 , (y1+y2+y3)/3) and using distance formula solve the expressions on LHS and RHS All the best. AKASH GOYAL AskiitiansExpert-IIT Delhi Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation. Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian. Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar..
Dear Sohan
You can do it by coordinates and vectors both methods
vector method is simple and involves less variables
assume o as origin
Position vectors of A,B,C as a,b,c respectively
OA=a, OB=b,OC=c
G=a+b+c/3
GA=a-(a+b+c)/3=2a-b-c/3
similarly you can write for GB,GC,GO
GA2=(4a2+b2+c2-4a.b+b.c-4a.c)/9
similarly write for others and just calculate the expression
similarly using coordinate
you can take o as (0,0)and A,B,C as (x1,y1),(x2,y2)and (x3,y3)
G will be ((x1+x2+x3)/3 , (y1+y2+y3)/3)
and using distance formula solve the expressions on LHS and RHS
All the best.
AKASH GOYAL
AskiitiansExpert-IIT Delhi
Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.
Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.
Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar..
if G is the centroid of the triangle ABC, prove by vector method thatGA+GB+GC=0
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