Question

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

Answer 1

We know that, 
The median of a triangle divides the triangle into two triangles of equal area.

AD is the median of ΔABC
Therefore, 
ar(ΔADB) =1/2 ar(ΔABC)    ............(1)

BE is the median of ΔABD
Therefore, 
ar(ΔBED) =1/2 ar(ΔADB)    
But
ar(ΔADB) =1/2 ar(ΔABC)    (from equation (1))

Therefore,

ar(ΔBED) =1/2[1/2 ar(ΔABC)]

ar(ΔBED) =1/4 ar(ΔABC)

Answer 2

area of triangle ADB = 1/2 * altitude * DB = 1/2 * altitude  * DC
             = 1/2 * altitude * (BC/2 )  = 1/2 * area of triangle ABC

similarly, 
 area of triangle  BED = 1/2 * base DE * altitude from B
                 = 1/2 * base EA * altitude from B
                   = 1/2 * (AD/2) * altitude from B
                     = 1/2 * area of triangle ADB

area of triangle DEB = 1/4 * area of triangle ABC