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Answer 1

Question : BD || CA, E is mid point of CA and BD = ½ CA and ar(∆ABC) = 20 unit². Prove that ar(∆DBC) = 10 unit²

Given : BD || CA, E is mid point of CA and BD = ½ CA and ar(∆ABC) = 20 unit².

To prove : ar(∆DBC) = 10 unit²

Proof : BD || AC and BD = ½ AC

Since ½ AC = CE therefore,

BD || CE and BD = CE

So, BCED is || gm

Therefore,

ar(∆DBC) = ½ ar(||gm BCED)

and, ar(∆BCE) = ½ ar(||gm BCED)

Hence, ar(∆DBC) = ar(∆BCE) _(i)

In ∆ABC, BE is median so,

ar(∆BCE) = ar(∆AEB)

So, ar(∆BCE) = ½ ar(∆ABC) _(ii)

From (i) and (ii) we get,

ar(∆DBC) = ½ ar(∆ABC)

ar(∆DBC) = ½ × 20 unit²

ar(∆DBC) = 10 unit²

Q.E.D

Answer 2

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