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A B C D is a quadrilateral; P , Q , Ra n...

`A B C D` is a quadrilateral; `P , Q , Ra n dS` are the points of trisection of side `A B ,B C ,C Da n dD A` respectively and are adjacent to `Aa n dC` ; prove that `P Q R S` is parallelogram.

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GIVEN A quadilateral ABCD in which P,Q,R,S are the points of trisection of AB,BC, CD and DA respectively (as shown).

TO PROVE PQRS is paarallellorgam.
CONSRUCTION Join AC.
PROOF In `Delta ABC`, we have `(BP)/(PA)=(BQ)/(QC)=(2)/(1)`
`:. PQ||AC...` (i) [ by the converse of Thale's theorm]
Also, in `Delta DAC`, we have
`(DS)/(SA)=(DR)/(RC)=(2)/(1)`
`:. SR||AC......` (ii) [ by the converse of Thale's theorm]
Thus, `PA||SR` [ from (i) and (ii)]
Similarly, by joining BD we can prove that `SP||RQ`.
Hence, PQRS is a parallelogram.
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