Question

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Q- In the given figure , BD || CA , E is the mid point of CA and BD = 1/2 of CA .
Prove that :- ar. (∆ABC) = 2 ar.(∆DBC)

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Answer 1

Answer:

we can easily prove using formula because base and height are same in both formulas

Answer 2

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BD||CA_________(GIVEN)

BD||CE_________(CE is part of CA)

AC/2=BD________(GIVEN)____(i)

E is Mid Point of AC__(GIVEN)

CE=AE & AC/2=CE____(ii)

From (i) and(ii):-

BD=CE

BD||CE

As one pair of opposite sides is equal and parallel so:-

BCED=Parallelogram

ar.(∆CDB)=ar.(∆BEC) ___(These ∆s are on the same base BC & btw. the same parallels BC&ED)

_________(iii)

In ∆ABC:-

E is the Mid Point of AC.AE=AC.

BE=Median_______(AE=CE)

As median divides the ∆ into 2 ∆s of equal areas.

ar.(∆ABE)=ar.(∆CBE)______(iv)

ar(ABE)+ar(CBE)=ar(ABC)

2ar(BEC)=ar(ABC)_________(v)

From (iii)&(v):-

2ar(∆DBC)=ar(∆ABC)

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