A, B, C, D are the mid-points of the sides PQ, QR, RS and SP of a parallelogram PQRS respectively. SA, SB, QC and QD have been joined to intersect at E and F. Show that EQFS is a parallelogram.

Question

A, B, C, D are the mid-points ​of the sides PQ, QR, RS and SP
of a parallelogram PQRS respectively SA, SB, QC and QD have been
joined to intersect at E and F Show that EQFS is a
parallelogram

Neha Sethi · Accepted Answer

Dear student

Since A and C are the mid points of PQ and SR respectively
So, AP=AQ=PQ2  1SC=CR=SR2 
2and PQ=SR   
3    opposite sides of  a ∥gm are equal So, from 1,2 and 3AP=AQ=SC=CR 
 4Now In △APS and △CRQ AP=CR    using 4 ∠P=∠R    Opp angles of  a ∥gm are equalPS=RQ       opposite sides of  a ∥gm are equal So, △APS=△CRQ     SAS Rule⇒AS=CQ       C
P C
TSince AS=CQ and SC=AQ⇒ASCQ is a ∥gm      opposite sides are equal  Similarly DSBQ is a  ∥gmSince ASCQ is a ∥gmSo, AS∥CQ⇒ES∥FQ   
5  opposite sides of  a ∥gm are ∥     Also, DSBQ is a ∥gm So, DQ∥SB⇒EQ∥SF  
6 opposite sides of  a ∥gm are ∥ So, by 5 and 6ESFQ is a ∥gm
Regards

A, B, C, D are the mid-points ​of the sides PQ, QR, RS and SP of a parallelogram PQRS respectively. SA, SB, QC and QD have been joined to intersect at E and F. Show that EQFS is a parallelogram.

A, B, C, D are the mid-points of the sides PQ, QR, RS and SP of a parallelogram PQRS respectively. SA, SB, QC and QD have been joined to intersect at E and F. Show that EQFS is a parallelogram.