Question

In the given fig ABCD is a quadrilateral P, Q,R,and S are the points of trisection of the sides of AB,BC,CD< and DA, respectively Prove that PQRS is a parallelogram

Answer 1

ABCD is a quadrilateral in which point P,Q,R,S are the points of trisection of the sieds of AB, BC,CD and DA respectively
We have to prove that PQRS is a parallelogram
Join AC
PROOF : - since P,Q,R,S are the points of trisection of sides AB,BC,CD and DA respectively, so BP =2 PA, BQ=2QC, DR=2RC, DS=2SA
$\frac{BP}{PA}=2$
$\frac{BQ}{QC}=2$
$\frac{DR}{RC}=2$
$\frac{DS}{SA}=2$
From (i) and (ii) we have $\frac{BP}{PA}=\frac{BQ}{QC}$
$\Rightarrow PQ\parallel AC$ ....(v) ( Using converse of Thales Theorem)
From (iii) and (iv) we have $\frac{DR}{RC}=\frac{DS}{SA}$
$\Rightarrow RS\parallel AC$,.......(iv)
(Using converse of Thales Theorem)
From (v) and (vi) we get
$PQ\parallel SR$
Similarly $PS\parallel RQ$
Hence PQRS is a parallelogram