Math, asked by Atharvadharmik, 1 year ago

PQRS is a parallelogram and AB||PS then prove that OC||SR

Answers

Answered by palsabita1957
15

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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