Question

A point D is taken on the side BC of a ΔABC such that BD = 2DC. Prove that ar(Δ ABD) =
2ar (ΔADC).

Answer 1

GIven that , 

In ΔABC, BD = 2 DC

To prove: ar ( ΔABD ) = 2ar (ΔADC)
Construction: Take a point E on BD such that BE = ED 
Proof : Since, BE = ED and 2 BD = 2DC 
Then, BE = ED = DC

We know that median of Δ^ledivides it into two equal   Δ^les 
∴ In , ΔABD , AE  is a median

Then, area (ΔABD) 2ar (ΔAED)  .....(1)
In , ΔAEC , AD is a median
Then area (ΔAED) =  area  (ΔADC)   ...... (2)

Compare equation (1) and (2)
Area (ΔABD)  = 2ar (ΔADC).