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Question

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ=2OPQ


Solution


We know that, the lengths of tangents drawn from an external point to a circle are equal.
TP=TQ
In ΔTPQ
TP = TQ
TQP=TPQ(1) (In a triangle, equal sides have equal angles opposite to them)
TQP+TPQ+PTQ=180 (Angle sum property)
2TPQ+PTQ=180 (Using (1))
PTQ=1802TPQ(1)
We know that, a tangent to a circle is perpendicular to the radius through the point of contact.
OPPTOPT=90OPQ+TPQ=90OPQ=90TPQ2OPQ=2(90TPQ)=1802TPQ(2)
From (1) and (2), we get
PTQ=2OPQ

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