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Byju's Answer
Standard VIII
Mathematics
The Centroid
If the median...
Question
If the median of a
△
A
B
C
intersect at G. show that ar (
△
A
G
C
) = ar (
△
A
G
B
) = ar (
△
B
G
C
) =
1
3
ar (
△
A
B
C
)
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Solution
AM , BN & CL are medians
to prove -ar. ΔAGB = ar.ΔAGC , ar.ΔAGB = ar. ΔBGC & ΔAGB= 1/3ar.ΔABC
proof ,
in ΔAGB & ΔAGC
AG is the median
∴ ar. ΔAGB = ar.ΔAGC
similarly ,
BG is the median
∴ ar.ΔAGB = ar. ΔBGC
so we can say that ar. ΔAGB = ar.ΔAGC = ΔBGC
now ,
ΔAGB + ΔAGC + ΔBGC = ar. ΔABC
1/3ΔAGB +1/3ΔAGC + 1/3ΔBGC ( they are equal in area ) = ar. ΔABC
so we can say that ,
ΔAGB = ar.ΔAGC = ΔBGC = ar 1/3 ΔABC
( PROVED )
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