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Question
P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.
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Solution
Given: In a parallelogram ABCD, P and Q are the mid-points of AS and CD, respectively.
To show: PRQS is a parallelogram.
Proof: Since, ABCD is a parallelogram.
AB || CD
⇒ AP || QC
Also, AB = DC
`1/2`AB = `1/2`DC ...[Dividing both sides by 2]
⇒ AP = QC ...[Since, P and Q are the mid-points of AB and DC]
Now, AP || QC and AP = QC
Thus, APCQ is a parallelogram.
∴ AQ || PC or SQ || PR ...(i)
Again, AB || DC or BP || DQ
Also, AB = DC
⇒ `1/2`AB = `1/2`DC ...[Dividing both sides by 2]
⇒ BP = QD ...[Since, P and Q are the mid-points of AB and DC]
Now, BP || QD and BP = QD
So, BPDQ is a parallelogram.
∴ PD || BQ or PS || QR ...(ii)
From equations (i) and (ii),
SQ || RP and PS || QR
So, PRQS is a parallelogram.
Hence proved.
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