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ABC is a triangle in which D is the mid-point of BC and E is the mid-point of AD. Prove that the area of ∆BED= 1/4 area of ∆ABC. 

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Given: A ∆ABC in which D is the mid-point of BC and E is the mid-point of AD. 

To prove: ar(∆BED) = 1/4 ar(∆ABC). 

Proof : ∵AD is a median of ∆ABC. 

∴ ar(∆ABD) = ar(∆ADC) = 1/2 ar(∆ABC) .....(i) 

 [∴Median of a triangle divides it into two triangles of equal area) = 1/2 ar(∆ABC) Again, 

∵ BE is a median of ∆ABD, 

∴ ar(∆BEA) = ar(∆BED) = 1/2 ar(∆ABD) 

 [∴Median of a triangle divides it into two triangles of equal area] 

And 1/2 ar(∆ABD) = 1/2 x 1/2 × ar(∆ABC) [From (i)] 

∴ ar(∆BED) = 1/4 ar(∆ABC). 

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