Question

Prove that the parallelograms on the same base and between the same parallels are equal in area.

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Solution

Let OACB and OAED be two parallelograms on the same base OA and between the same parallels are equal in area.
Let

OA with rightwards arrow on top space equals space straight a with rightwards arrow on top space space and space OB with rightwards arrow on top space equals space straight b with rightwards arrow on top

Since BD is parallel to OA,

therefore space space space space space space BD with rightwards arrow on top space equals space straight x straight a with rightwards arrow on top comma space where space straight x space is space some space scalar. therefore space space space space space space space OD with rightwards arrow on top space equals space OB with rightwards arrow on top space plus space BD with rightwards arrow on top space equals space straight b with rightwards arrow on top space plus space straight x space straight a with rightwards arrow on top

Now vector area of parallelogram OAED =

OA with rightwards arrow on top space cross times space OD with rightwards arrow on top

equals space straight a with rightwards arrow on top space cross times space left parenthesis straight b with rightwards arrow on top space plus space straight x space straight a with rightwards arrow on top right parenthesis space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight x space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space 0 with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top close square brackets space equals space OA with rightwards arrow on top space cross times space OB with rightwards arrow on top space equals space vector space area space of space parallelogram space OACB.

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