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Diagonals AC and BD of a quadrilatera...

Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that `a r"\ "(A O D)"\ "="\ "a r"\ "(B O C)dot` Prove that ABCD is a trapezium.

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Given ABCD is a quadrilateral and diagonal AC and BD intersect atO.
Also, `ar(DeltaAOD) = ar(DeltaBOC)`
On adding both sides `ar(DeltaAOB)`, we get
`ar(DeltaAOD) + ar(DeltaAOB) = ar(DeltaBOC) + ar(DeltaAOB)`
`rArr ar(Delta ADB) = ar(Delta ACB)`
Now, `Delta ACB and Delta ADB` lie on same base AB
and `ar(DeltaADB) = ar(DeltaACB)`
Hence, `DeltaACB and Delta ADB` lie between same parallel lines.
`:. AB ||DC` ltrbgt Hence, ABCD is a trapezium.
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