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Question

In a triangle ABC, E is the midpoint of median AD, show that areaABE=areaACE
569732_7587bbbe27824643873166057a9e9d11.png

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Solution

AD is the median of ΔABC.
Therefore, it will divide ΔABC into two triangles of equal area.
Area ΔABD = Area ΔACD
area(ΔABD)=12area(ΔABC) ------------(1)
In ΔABD, E is the mid-point of AD.
Therefore, BE is the median.
Then Area ΔBED = Area ΔABE
area(ΔABE)=12area(ΔABD) ------------(2)
area(ΔABE)=12×12area(ΔABC)
area(ΔABE)=14area(ΔABC)...........................(3)
In ΔADC, E is the mid-point of AD.
Therefore, EC is the median.
Area ΔAEC = Area ΔCED
area(ΔACE)=12area(ΔADC)
area(ΔACE)=12×12area(ΔABC)
area(ΔACE)=14area(ΔABC)....................(4)
From (3) and (4) we get
area(ΔABE)=area(ΔACE)




711649_569732_ans_146d59f522e746b5a4edbdcd3149381e.png

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