Diagonals AC & BD of a quadilateral ABCD intersect at O such that area AOD=area BOC. Prove ABCD is a trapezium

Answer 1

Add the area of triangle DOC to both AOD and BOC,
Since area of AOD = BOC
=> area(ADC) = area(BDC) [ the triangles obtained by the previous additions ]
Since the base DC is common to both, and the areas are equal
=> Height from A to DC in triangle ADC = Height from B to DC in triangle BDC
Thus the points A and B are equidistant from DC, making AB || DC and the given quadrilateral is a trapezium

Answer 2

ABCD is a quadtilateral.
Its diagonals AC and BD intersect each other at point O so that the area of triangle AOD is equal to that of triangle BOC.
To prove that ABCD is trapezium
Triangle AOD=triangle BOC [Given}
Adding area of triangle AOB to both sides
Triangle AOD+triangle AOB =Triangle BOC+triangle AOB
So, triangle ABD=Triangle ABC
But these two triangles are on the same base AB and on the same side of AB
Therefore,they must be between same pair of parallel lines
Or,AB||CD
Or,ABCD is a trapezium (Proved)

Answer 3

ABCD is a quadtilateral.
Its diagonals AC and BD intersect each other at point O so that the area of triangle AOD is equal to that of triangle BOC.
To prove that ABCD is trapezium
Triangle AOD=triangle BOC [Given}
Adding area of triangle AOB to both sides
Triangle AOD+triangle AOB =Triangle BOC+triangle AOB
So, triangle ABD=Triangle ABC
But these two triangles are on the same base AB and on the same side of AB
Therefore,they must be between same pair of parallel lines
Or,AB||CD
Or,ABCD is a trapezium (Proved)

Answer 4

ar(AOD) = ar(BOC)
This can be solved by area additions.

Add ar(COD) to both the sides
ar(AOD) + ar(COD) = ar(BOC) + ar(COD) This gives:
ar(ACD) = ar(BCD)
Triangles ACD and BCD have same base CD and equal areas.
So, they lie between the same parallels.
This implies that AB is parallel to CD

(i.e.) ABCD is a trapezium

Answer 5

The Point O is somewhere above ABCD, and the 2 triangles AOD and BOC are pointing north east and north west with their bases.
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Answer 6

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Answer 7

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