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Question
Prove that the diagonals of a parallelogram divide it into four triangles of equal area.
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Solution
The diagonals of a parallelogram bisect each other.
Therefore, O is the mid-point of AC and BD.
Bo is the median in ΔABC.
Therefore, it will divide it into two triangles of equal areas
∴ ar(ΔAOB) = ar(ΔBOC)......(i)
In ΔBCD, CO is the median.
∴ ar(ΔBOC) = ar(ΔCOD).......(ii)
Similarly, ar(ΔCOD) = ar(ΔAOD)......(iii)
From (i), (ii) and (iii)
ar(ΔAOB) = ar(ΔBOC) = ar(ΔCOD) = ar(ΔAOD)
Hence, diagonals of a parallelogram divide it into foor triangles of equal areas.
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