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

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#### Solution

AD is the median of ΔABC. Therefore, it will divide ΔABC into two triangles of equal areas.

∴ Area (ΔABD) = Area (ΔACD)

⇒ Area (ΔABD) = 1/2Area (ΔABC)... (1)

In ΔABD, E is the mid-point of AD. Therefore, BE is the median.

∴ Area (ΔBED) = Area (ΔABE)

⇒ Area (ΔBED) = 1/2Area (ΔABD)

⇒ Area (ΔBED) = 1/2*1/2Area (ΔABC) [From equation (1)]

⇒ Area (ΔBED) = 1/4Area (ΔABC)

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