In a triangle ABC, E is the mid-point of median AD. Show that ar(BED) = ¼ ar(ABC).

Solution:

ar(BED) = (1/2)×BD×DE

Since, E is the mid-point of AD,

AE = DE

Since, AD is the median on side BC of triangle ABC,

BD = DC

,

DE = (1/2) AD — (i)

BD = (1/2)BC — (ii)

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

ar(BED) = (1/2)×(1/2)BC × (1/2)AD

⇒  ar(BED) = (1/2)×(1/2)ar(ABC)

⇒ ar(BED) = ¼ ar(ABC)