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Areas of Parallelograms and Triangles
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15/07/2021 1:16 pm
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Show that the diagonals of a parallelogram divide it into four triangles of equal area.
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15/07/2021 1:18 pm
O is the mid point of AC and BD. (diagonals of bisect each other)
In ΔABC, BO is the median.
∴ ar(AOB) = ar(BOC) — (i)
In ΔBCD, CO is the median.
∴ ar(BOC) = ar(COD) — (ii)
In ΔACD, OD is the median.
∴ ar(AOD) = ar(COD) — (iii)
In ΔABD, AO is the median.
∴ ar(AOD) = ar(AOB) — (iv)
From equations (i), (ii), (iii) and (iv), we get,
ar(BOC) = ar(COD) = ar(AOD) = ar(AOB)
Hence, we get, the diagonals of a parallelogram divide it into four triangles of equal area.