# ABCD is a parallelogram whose diagonals intersect at O. If P is any point on BO, prove that(i) ar(Δ ADO) = ar(Δ CDO)(ii) ar(Δ ABP) = ar(Δ CBP)

Given that,

ABCD is a parallelogram

To prove:

(i) Area ( ) = Area ( )

(ii) Area ( ) = Area ( )

Proof: We know that,

Diagonals of a parallelogram bisect each other

Therefore,

AO = OC and,

BO = OD (i) In ΔDAC, DO is a median.

Therefore,

Hence, proved

(ii) In , since BO is a median

Then,

Area (ΔBAO) = Area (ΔBCO) (i)

In a ΔPAC, since PO is the median

Then,

Area (ΔPAO) = Area (ΔPCO) (ii)

Subtract (ii) from (i), we get

Area (ΔBAO) - Area (ΔPAO) = Area (ΔBCO) - Area (ΔPCO)

Area (ΔABP) = Area (ΔCBP)

Hence, proved

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