points D E and F are the midpoints of sides of triangle ABC point G is the centroid of triangle ABC show that. G is the centroid of triangle d e s also
i want correct answer or reported
Answers
G is centroid of Triangle DEF also , if G is centroid of triangle ABC & D , E , F are mid points of sides of triangle
Step-by-step explanation:
Let say Coordinates of A , B & C are
( Ax Ay) , (Bx , By) & (Cx , Cy) respectively
Then Centriod G - coordinates = ( Ax + Bx + Cx)/3 , (Ay + By + Cy)/3
D , E & F are mid points of AB , BC & AC
D = (Ax + Bx)/2 , (Ay + By)/2
E = (Bx + Cx)/2 , (By + Cy)/2
F = (Ax + Cx)/2 , (Ay + Cy)/2
Centroid of triangle DEF
= ( (Ax + Bx)/2 + (Bx + Cx)/2 + (Ax + Cx)/2) / 3 , ( (Ay + By)/2 + (By + Cy)/2 + (Ay + Cy)/2) / 3
= ( Ax + Bx + Bx + Cx + Ax + Cx)/ 6 , ( Ay + By + By + Cy + Ay + Cy)/ 6
= 2(Ax + Bx + Cx)/6 , 2(Ay + By + Cy)/6
= (Ax + Bx + Cx)/3 , (Ay + By + Cy)/3
= Same as coordinates of G
Hence G is centroid of Triangle DEF also
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