P and Q are the mid-points of the opposite sides AB and CD of parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.


Given, P and Q are mid-points of AB and CD.


Now, AB||CD,


AP||QC


Also, AB = DC


AB = DC


AP = QC


Now, AP||QC and AP = QC


APCQ is a parallelogram.


AQ||PC or SQ||PR


Again,


AB||DC AB = DC


BP = QD


Now, BP||QD and BP = QD


BPDQ is a parallelogram


So, PD||BQ or PS||QR


Thus, SQ||RP and PS||QR


PQRS is a parallelogram.


7