The diagonals of quad. ABCD intersect at O .Prove that if BO = OD, then ar (triangle ABC) = ar (triangle ADC).

given: ABCD is a quadrilateral. diagonals AC and BD intersect at O. and BO=DO

TPT: area(ΔABC)=area(ΔADC)

construction: draw the perpendiculars AE and CF from A and C, such that 

AE⊥BD and CF⊥BD

proof:

area(ΔBAO)=

area(ΔBAO)=area(ΔADO)...........(1)

area(ΔBOC)=

area(ΔBOC)=area(ΔDOC)............(2)

adding (1) and (2)

area(ΔBAO)+area(ΔBOC)=area(ΔADO)+area(ΔDOC)

area(ΔABC)=area(ΔADC)

which is the required result.

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