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Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.

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We have,

Given:

There are more than 1 parallelogram, and their bases can be taken as common and they are between same parallels.

To Prove:

These parallelograms whose bases are same and are between the same parallel sides have equal area.

Proof:

Let ABCD and ABFE be two parallelograms on the same base AB and between same parallel lines AB and DF.

Here,

AB ∥ DC and AE ∥ BF

We can represent area of parallelogram ABCD as,

[∵ a scalar term can be taken out of a vector product]

From equation (i) and (ii), we can conclude that

Area of parallelogram ABCD = Area of parallelogram ABFE

Thus, parallelogram on same base and between same parallels are equal in area.

Hence, proved.

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