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Question
ABC is a triangle whose vertices are A(-4, 2), B(O, 2) and C(-2, -4). D. E and Fare the midpoint of the sides BC, CA and AB respectively. Prove that the centroid of the Δ ABC coincides with the centroid of the Δ DEF.
Solution
Let D , E and F be the midpoints of the sides AB , AC and BC of Δ ABC respectively.
∴ AD : DB = BF : FC = AE : EC = 1 : 1
Coordinates of D are ,
D (x , y) = D `((0 - 4)/2 , (2 + 2)/3) = "D" (-2 , 2)`
Similarly ,
E (a , b) = E `((-4 - 2)/2 , (2 - 4)/2)` = E (-3 , -1)
and ,
F (p,q) = F `((0 - 2)/2 , (2 - 4)/2)` = F (-1 , -1)
Coordinates of centroid of Δ ABC are ,
`= ((-4-2+0)/3 , (2 - 4 + 2)/3)` = (-2 , 0)
Coordinates of centroid of Δ DEF are ,
`= ((-2-3-1)/3 , (2 - 1 - 1)/3) = (-2 , 0)`
Thud the centroid of Δ DEF coincides with centroid of Δ DEF.
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