1.

If each diagonal of a quadri lateral dividesit into two triangles of equal areas, thenprove that quadrilateral is a paralldogram,​

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Answer:

A QUADRILATERAL ABCD and its diagonals AC and BD divides it into two TRIANGLES of equal areas i.e. AR(△ABC)=ar(△CDB)

and ar(△ABC)=ar(△ACD)

Proof :

⇒ ar(△ABC)=ar(△ACD) ------ ( 1 )

⇒ ar(△ABC)+ar(△ACD)=ar(□ABCD) ----- ( 2 )

From ( 1 ) and ( 2 )

⇒ 2ar(△ABC)=ar(□ABCD) ----- ( 3 )

⇒ and ar(△ABD)=ar(△BCD) ----- ( 4 )

⇒ ar(△ABD)+ar(△BCD)=ar(□ABCD) ---- ( 5 )

From ( 4 ) and ( 5 )

⇒ 2ar(△ABD)=ar(□ABCD) ---- ( 6 )

From ( 3 ) and ( 6 ) we get,

⇒ 2ar(△ABC)=2ar(△ABD)

⇒ ar(△ABC)=ar(△ABD)

SINCE, △ABC and △ABD are on the some base AB. Therefore, they must have equal corresponding altitudes.

i.e. Altitude from C of △ABC= Altitude from D of △ABD

⇒ DC∥AB

Similarly AD∥BC

Hence ABCD is a parallelogram.


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