Prove that the area of a triangle is half the product of any its sides and the corresponding altitudes .

Construction: Trough C and A draw CP parallel to BA and AP parallel to BC, intersecting each other at P.


CP parallel to BA  (By Construction)

AP parallel to BC  (By Construction)

Therefore BCPA is a parallelogram.

Hence AC is a diagonal of parallelogram BCPA.

Therefore, area of triangle = ½ ar(IIgm BCPA) = ½ × BC × AD  (As BC is the base and AD is the altitude of IIgm BCPA).


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