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.

Name: 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.
Uploaded: Thu, 2021-01-07 23:56
Description: 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.

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.

Answers (1)

Solution.
Given: ABCD is a parallelogram, P and Q are the mid-points of AB and CD
To Prove: PRQS is a parallelogram.
Proof: Since ABCD is a parallelogram $AB\parallel CD \Rightarrow AP\parallel QC$

Also, AB = DC …..(1)
$\frac{1}{2}AB=\frac{1}{2}DC$ {dividing by 2}
$AP = QC$
{ $\because$ P and Q are the mid-points of AB and DC}
As, $AP = QC$ and   $AP\parallel QC$
Thus APCQ is a parallelogram
$AQ\parallel PC$   or $SQ\parallel PR$     …..(2)
$AB\parallel CD$ or $BP\parallel DQ$
$AB=DC$
$\frac{1}{2}AB=\frac{1}{2}DC$ {dividing both sides by 2}
$BP=QD$    { $\because$ P and Q are the mid-points of AB and DC}
$BP\parallel QD$ and $BQ\parallel PD$
So, BPDQ is a parallelogram
$PD\parallel BQ$ or $PS\parallel QR$    …..(3)
From equation 2 and 3
$SQ\parallel RP$   and $PS\parallel QR$
So, PRQS is a parallelogram.
Hence Proved