if each diagonal of a quadrilateral separate it into two Triangles of equal area then show that quadrilateral is a parallelogram.
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Answered by
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In quadrilateral ABCD, AC is a diagonal.
∴ ar = ar
ar + ar = ar + ar
In quadrilateral ABCD, BD is diagonal
∴ ar = ar
ar + ar = ar + ar
From equation (i) and (ii),
We have ;
ar - ar = ar - ar
So,
2ar = 2ar
=
ar + ar = ar + ar
ar = ar
and having common base AB and line between two lines AB and DC.
∴ AB || DC
Similarly we can prove that AD || BC.
∴ ABCD is a parallelogram.
∴ ar = ar
ar + ar = ar + ar
In quadrilateral ABCD, BD is diagonal
∴ ar = ar
ar + ar = ar + ar
From equation (i) and (ii),
We have ;
ar - ar = ar - ar
So,
2ar = 2ar
=
ar + ar = ar + ar
ar = ar
and having common base AB and line between two lines AB and DC.
∴ AB || DC
Similarly we can prove that AD || BC.
∴ ABCD is a parallelogram.
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