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Question

In the given figure two tangents PQ and PR are drawn to a circle with centre O from an external point P. Prove that QPR=2OQR
1332940_21afebfc1739460cbc8e6da7f6dd3aa5.png

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Solution

Given that PQ and QR are two tangents drawn to a circle with centre O from an external point P.

To prove: QPR=2OQR

Construction: Join QR, OQ and OR.

Proof: We know that lengths of a tangent drawn from an external point to a circle are equal.

PQ=QR

ΔPQR is an isosceles triangle

PQR=PRQ

In ΔPQR

PQR+PRQ+QPR=180o

PQR+PQR+QPR=180o

2.PQR=180oQPR

PQR=12(180oQPR)

PQR=90o12QPR

12QPR=90oPQR …………(1)

Since PQ is perpendicular to PQ.

OQP=90o

OQR+PQR=90o

OQR=90PQR ………..(2)

OQR=12OQR

2.OQR=QPR

QPR=2OQR

Hence, it proved.

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