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Q.

Two tangents TP and TQ are drawn to a circle with center O from an external point T. Then ∠PTQ= ____ ∠OPQ.


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Detailed Solution

TP = TQ …(1)
⇒∠TQP=∠TPQ …(2)
OP is perpendicular to TP
⇒∠OPT= 90∘
⇒∠OPQ + ∠TPQ= 90∘
⇒∠TPQ= 90∘−∠OPQ …(3)
In triangle PTQ,
⇒∠TPQ + ∠PQT + ∠QTP = 180∘
90∘−∠OPQ +∠TPQ + ∠QTP = 180∘
⇒ 2(90∘−∠OPQ) + ∠QTP = 180∘
⇒ 180∘−2∠OPQ + ∠PTQ = 180∘
⇒ 2∠OPQ = ∠PTQ
⇒ 2(90∘− ∠OPQ) + ∠QTP = 180∘
⇒ 180∘− 2∠OPQ + ∠PTQ = 180∘
⇒ 2∠OPQ = ∠PTQ
Two tangents TP and TQ are drawn to a circle with center O from an external point T. Then ∠PTQ= 2∠OPQ.
 
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Two tangents TP and TQ are drawn to a circle with center O from an external point T. Then ∠PTQ= ____ ∠OPQ.