Through A, B and C lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a `Delta`ABC as shown in figure. Show that `BC=(1)/(2)QR`.

Explanation

Given In `Delta`ABC, PQ `||` AB and PR `||` AC and RQ `||` BC.
To show ` BC = (1)/(2)QR`
Proof In qudrilateral BCAR, `BR||CA and BC||RA`
So, quadrilateral, BCAR is a parallelogram.
`therefore" "BC=AR" "...(i)`
Now, in quadrilateral BCQA, BC`||`AQ
and `" "AB||QC`
So, quadrilateral BCQA is a parallelogram,
`therefore " " BC = AQ " "...(ii)`
On adding Eqs. (i) and (ii), we get
2BC=AR + AQ
`rArr" "2BC=RQ`
`rArr" "BC=(1)/(2)QR`
Hence proved.