Q. 133.7( 3 Votes )

A point D is taken on the side BC of a Δ ABC such that BD = 2DC. Prove that

ar(Δ ABD) = 2ar(Δ ADC)

Answer :

Given that,

In Δ ABC,


We have


BD = 2DC


To prove: Area () = 2 Area ()


Construction: Take a point E on BD such that, BE = ED


Proof: Since,


BE = ED and,


BD = 2DC


Then,


BE = ED = DC


Median of the triangle divides it into two equal triangles


Since,


AE and AD are the medians of ΔABD and AEC respectively


Therefore,


Area (ΔABD) = 2 Area (ΔAED) (i)


And,


Area (ΔADC) = Area (ΔAED) (ii)


Comparing (i) and (ii), we get


Area (ΔABD) = 2 Area (ΔADC)


Hence, proved



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