Mathematics
If each diagonal of a quadrilateral divides it into two triangles of equal areas, then prove that the quadrilateral is a parallelogram.
Theorems on Area
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Answer
Let ABCD be a quadrilateral such that each diagonal divides it into triangles of equal areas, then
area of △ABC = Area of ABCD, …….(1)
area of △ABD = Area of ABCD, ……(2)
area of △BCD = Area of ABCD, …….(3)
We know that,
Triangles on the same base and having equal areas lie between the same parallel lines.
From (1) and (2) we get,
Area of △ABC = Area of △ABD.
Since, △ABC and △ABD lie on same base AB and have equal area.
So, AB || CD.
From (1) and (3) we get,
∴ Area of △ABC = Area of △BCD.
Since, △ABC and △BCD lie on same base BC and have equal area.
So, BC || AD.
Since, AB || CD and BC || AD.
Hence, proved ABCD is a parallelogram.
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