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D,E and F are the midpoints of sides BC,CA and AB respetiely of triangle ABC and G is the centroid of the triangle,then GD+GE+GF=
Asked by g_archanasharma | 15 Apr, 2019, 17:51: PM
answered-by-expert Expert Answer
L e t space A left parenthesis x subscript 1 comma y subscript 1 right parenthesis comma space B left parenthesis x subscript 2 comma y subscript 2 right parenthesis space a n d space C left parenthesis x subscript 3 comma y subscript 3 right parenthesis space b e space t h e space c o o r d i n a t e s space o f space v e r t i c e s space o f space t r i a n g l e space A B C G space i s space t h e space c e n t r o i d comma space t h e n G equals open parentheses fraction numerator x subscript 1 plus x subscript 2 plus x subscript 3 over denominator 3 end fraction comma fraction numerator y subscript 1 plus y subscript 2 plus y subscript 3 over denominator 3 end fraction close parentheses A l s o comma space D equals open parentheses fraction numerator x subscript 2 plus x subscript 3 over denominator 2 end fraction comma fraction numerator y subscript 2 plus y subscript 3 over denominator 2 end fraction close parentheses comma space E equals open parentheses fraction numerator x subscript 1 plus x subscript 3 over denominator 2 end fraction comma fraction numerator y subscript 1 plus y subscript 3 over denominator 2 end fraction close parentheses space a n d space F open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction comma fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses N o w comma space G D equals square root of open parentheses fraction numerator x subscript 2 plus x subscript 3 over denominator 2 end fraction minus fraction numerator x subscript 1 plus x subscript 2 plus x subscript 3 over denominator 3 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 2 plus y subscript 3 over denominator 2 end fraction minus fraction numerator y subscript 1 plus y subscript 2 plus y subscript 3 over denominator 3 end fraction close parentheses squared end root space space space space space space space space space space space space space space equals square root of open parentheses fraction numerator x subscript 2 plus x subscript 3 minus 2 x subscript 1 over denominator 6 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 2 plus y subscript 3 minus 2 y subscript 1 over denominator 6 end fraction close parentheses squared end root S i m i l a r l y comma space G E equals square root of open parentheses fraction numerator x subscript 1 plus x subscript 3 minus 2 x subscript 2 over denominator 6 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 1 plus y subscript 3 minus 2 y subscript 2 over denominator 6 end fraction close parentheses squared end root space a n d space G F equals square root of open parentheses fraction numerator x subscript 1 plus x subscript 2 minus 2 x subscript 3 over denominator 6 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 1 plus y subscript 2 minus 2 y subscript 3 over denominator 6 end fraction close parentheses squared end root S o space w e space h a v e G D plus G E plus G F equals square root of open parentheses fraction numerator x subscript 2 plus x subscript 3 minus 2 x subscript 1 over denominator 6 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 2 plus y subscript 3 minus 2 y subscript 1 over denominator 6 end fraction close parentheses squared end root plus square root of open parentheses fraction numerator x subscript 1 plus x subscript 3 minus 2 x subscript 2 over denominator 6 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 1 plus y subscript 3 minus 2 y subscript 2 over denominator 6 end fraction close parentheses squared end root plus square root of open parentheses fraction numerator x subscript 1 plus x subscript 2 minus 2 x subscript 3 over denominator 6 end fraction close parentheses squared plus open parentheses fraction numerator y subscript 1 plus y subscript 2 minus 2 y subscript 3 over denominator 6 end fraction close parentheses squared end root space space space space space space space space space space space space space space space space space space space space space space equals 1 over 6 open parentheses square root of open parentheses x subscript 2 plus x subscript 3 minus 2 x subscript 1 close parentheses squared plus open parentheses y subscript 2 plus y subscript 3 minus 2 y subscript 1 close parentheses squared end root space plus square root of open parentheses x subscript 1 plus x subscript 3 minus 2 x subscript 2 close parentheses squared plus open parentheses y subscript 1 plus y subscript 3 minus 2 y subscript 2 close parentheses squared end root space plus square root of open parentheses x subscript 1 plus x subscript 2 minus 2 x subscript 3 close parentheses squared plus open parentheses y subscript 1 plus y subscript 2 minus 2 y subscript 3 close parentheses squared end root space close parentheses
Answered by Renu Varma | 16 Apr, 2019, 09:58: AM

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