Question

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

Answer 1

Given In ΔABC, PQ || AB and PR || AC and RQ || BC.

To show BC = `1/2` QR

Proof In quadrilateral BCAR, BR || CA and BC || RA

So, quadrilateral, BCAR is a parallelogram.

BC = AR  ...(i)

Now, in quadrilateral BCQA,

BC || AQ

And AB || QC

So, quadrilateral BCQA is a parallelogram,

BC = AQ  ...(ii)

On adding equations (i) and (ii), we get

2BC = AR + AQ

⇒ 2BC = RQ

⇒ BC = `1/2` QR

Now, BEDF is a quadrilateral, in which ∠BED = ∠BFD = 90°

∠FSE = 360° – (∠FDE + ∠BED + ∠BFD)

= 360° – (60° + 90° + 90°)

= 360° – 240°

= 120°