PQRS is a parallelogram and AB||PS then prove that OC||SR
Answers
This can be proved by applying similarity of triangles and converse of Thales Theorem .
Step-by-step explanation:
To prove - OC║SR
Proof - In ΔOPS and ΔOAB
∠POS = ∠AOB (common in both)
∠OSP = ∠OBA (corresponding angles are equal as PS║AB)
=> ΔOPS ~ ΔOAB [AA criteria]
=> PS/AB = OS/OB ........................(1) (sides in similar triangles are proportional)
In ΔCAB and ΔCRQ
As, QR║AB
=> ∠QCR = ∠ACB (common)
=> ∠CBA = ∠CRQ (corresponding angles are equal)
=> ΔCAB ~ ΔCQR [AA criteria]
=> CR/CB = QR/AB (sides in similar triangles are proportional)
Also, PS = QR [ PQRS is parallelogram]
=> CR/CB = PS/AB ......................(2)
From (1) and (2)
=> OS/OB = CR/CB
=> OB/OS = CB/CR
Subtracting 1 from both sides
So, OB/OS - 1 = CB/CR - 1
=> (OB - OS)/OS = (CB - CR)/CR
=> BS/OS = BR/CR
By converse of Thales Theorem
=> OC║SR . Hence proved .
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