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
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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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