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प्रश्न
In figure, if PQRS is a parallelogram and AB || PS, then prove that OC || SR.
उत्तर
According to the question,
PQRS is a parallelogram,
Therefore, PQ || SR and PS || QR.
Also given, AB || PS.
To prove:
OC || SR
From ∆OPS and OAB,
PS || AB
∠POS = ∠AOB ...[Common angle]
∠OSP = ∠OBA ...[Corresponding angles]
∴ ∆OPS ∼ ∆OAB ...[By AAA similarity criteria]
Then,
Using basic proportionality theorem,
We get,
`("PS")/("AB") = ("OS")/("OB")` ...(i)
From ∆CQR and ∆CAB,
QR || PS || AB
∠QCR = ∠ACB ...[Common angle]
∠CRQ = ∠CBA ...[Corresponding angles]
∴ ∆CQR ∼ ∆CAB
Then, by basic proportionality theorem
`("QR")/("AB") = ("CR")/("CB")`
⇒ `("PS")/("AB") = ("CR")/("CB")` ...(ii) [PS ≅ QR Since, PQRS is a parallelogram]
From equations (i) and (ii),
`("OS")/("OB") = ("CR")/("CB")`
or
`("OB")/("OS") = ("CB")/("CR")`
Subtracting 1 from L.H.S and R.H.S, we get,
`("OB")/("OS") - 1 = ("CB")/("CR") - 1`
⇒ `("OB" - "OS")/("OS") = ("CB" - "CR")/("CR")`
⇒ `("BS")/("OS") = ("BR")/("CR")`
SR || OC ...[By converse of basic proportionality theorem]
Hence proved.
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