Question : PQRS is a parallelogram and AB||PS. Prove that OC||SR.
Doubt by Muskan
Solution :
Given :
PQRS is a parallelogram.
AB || PS
To Prove : OC || SR
Proof :
PQRS is a parallelogram (Given)
PQ || SR & PQ = SR (opposite sides of parallelogram are parallel and equal)
PS || QR & PS = QR (opposite sides of parallelogram are parallel and equal)
In ∆ OPS and ∆OAB
∠OPS = ∠OAB (Corresponding angle)
∠POS = ∠AOB (Common)
∆OPS ~ ∆OAB (by AA similarity)
PS/AB = OS/OB (by CPST) — (1)
PS||QR (Proved above)
PS||AB (Given)
⇒ QR||AB
In ∆CQR and ∆CAB
∠CQR = ∠CAB (Corresponding angle)
∠QCR = ∠ACB (Common)
∆CQR ~ ∆CAB (By AA Similarity)
QR/AB = CR/CB (By CPST)
PS/AB = CR/CB [∵QR=PS] — (2)
From (1) and (2)
OS/OB = CR/CB
OB/OS = CB/CR (Taking Reciprocal)
Subtracting both sides from 1
(OB/OS) - 1 = (CB/CR) - 1
(OB-OS)/OS = (CB-CR)/CR
BS/OS = BR/CR
∴ SR || OC (Converse of Pythagoras Theorem in ∆BOC)