in a triangle ABC,E is the midpoint of median AD. Show that area of BED = 1/4 area of ABC

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Varun.Rawat answered this

Ashiq Habib answered this

triangle BED=CED (median ED divides triangle ECB into triangles of equal area)

tringle BED=BEA (median BE divides triangle BAD into triangles of equal area)

triangle CEA=CED (median CE divides triangle CAD into triangles of equal area)

therefore, triangle BED = BEA = CED = CEA..........BED + BEA + CED + CEA = triangle ABC

'OR'

triangle BED + BED + BED + BED = ABC (since all triangles are equal to each other)

therfore 4 (BED) = ABC ........ BED = ABC divided by 4 which is ABC/4 'OR' 1/4 x ABC.. hence proved

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pooshna answered this

ar(ABD)=1/2ar(abc) (ad is a median)

ar(ABE)=1/2ar(ABD) (be is a median)

BUT ar (ABD)=1/2 ar(ABC) (from step 1)

therefore ar(BED)=1/2 into 1/2 ar(ABC)

i.e ar(bed)=1/2 into 1/2 ar(ABC)=1/4ar(ABC)

Jigar Bhavesh answered this

GIVEN : Let ABC be a triangle

whose median intersect side BC

and E is the mid- point

TO PROVE :area ( BED ) = 1/4 area ( ABC )

PROOF : Join BE

We know that median divides the triangle into two triangles of equal areas ,

= AD is the median of Triangle ABC

So, area ( ABD ) = area ( ACD ) = 1/2 area ( ABC ) ----------- ( 1 )

As E is the mid point , i.e : BE is the median of triangle ABD ,

similarly , area ( BED) = area ( AEB )

or , area ( BED) = 1/2 area ( ABD )

Now , putting the value of area ( ABD ) from (1 ) ,

= area ( BED)= 1/2 [ 1/2 area ( ABC ) ]

= area (BED) = 1/2 * 1/2 area ( ABC )

= area ( BED ) = 1/4 area ( ABC)