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In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = 1/4 ar(ABC).

Q : 2      In a triangle ABC, E is the mid-point of median AD. Show that     \small ar(BED)=\frac{1}{4}ar(ABC).
 

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M manish

We have a triangle ABC and  AD is a median. Join B and E.

Since the median divides the triangle into two triangles of equal area.
\therefore ar(\DeltaABD) = ar (\DeltaACD) = 1/2 ar(\DeltaABC)..............(i)
Now, in triangle \DeltaABD,
BE is the median [since E is the midpoint of AD]
\therefore ar (\DeltaBED) = 1/2 ar(\DeltaABD)........(ii)

From eq (i) and eq (ii), we get

ar (\DeltaBED) = 1/2 . (1/2 ar(ar (\DeltaABC))
ar (\DeltaBED)  = 1/4 .ar(\DeltaABC)

Hence proved.

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