Question

M and N are mid points of sides DC and AB respectively of parallelogram ABCD . show that ar BEC= ar MCBN , when itis given that BD is parallel to EC

Answer 1

hi friend.....
:)

Given-    ABCD is a parallelogram

To prove-  area (BEC) = area (MCBN)

construction- PRODUCE AB TO E AND

JOIN CE

Proof;-
since M and N are the midpoints of the sides AB and CD,

      ar (MCBN) = ar (MDAN)

or, ar (MCBN) = 1/2 ar (ABCD)              ...(i)

and ar 1/2ar(ABCD) = ar(triangleBCD)   ...(ii)    (because diagonal of a llgm divides it into 2 triangles of equal area)

FROM EQUATION 1 AND 2, WE HAVE
           AR (MCBN =BCD)   ...(iii)

NOW,

     AB ll DC
or, AE ll DC
or, BE ll DC ...(iv)
also,BD ll CE ..(v)

so, from equation (iv) and (v), BECD is a llgm, whose diagonal is BC

so BEC=BCD......(vi)
from( v) and( vi)
we get ar BEC =ar MCBN....

hense the proof..

hope this helps u...
:)