Question

diagonals AC and BD of a quadrilateral ABCD inresect at O in such a way that ar(AOD) =ar(BOC). Prove that ABCD is a trapezium

Answer 1

Given -

a quad. ABCD ,

ar( AOD ) = ar ( BOC ).

To prove- ABCD is a trapezium

Proof -

ar(ADC)= ar ( AOD ) +ar(DOC)

similarly

ar(BDC) = ar(BOC) + ar(DOC)

ar( AOD ) = ar ( BOC ).

DOC IS COMMON HERE

Therefore,ar(ADC) = ar(BDC)

These two triangles are on the same base DC and their areas are also equal .

Therefore, they lie between the same parallels AB and DC.

AB||DC

Therefore, ABCD is a parallelogram with one pair of opposite sides parallel.

Therefore, ABCD is a trapezium.

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