Solve these folliwibg questions

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ques6

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 = 180º  (Angle sum property)

∴  2 ∠TPQ  + ∠PTQ = 180º  (Using(1))

⇒ ∠PTQ = 180º – 2 ∠TPQ  ...(1)

We know that, a tangent to a circle is perpendicular to the radius through the point of contact.

OP ⊥ PT,

∴ ∠OPT = 90º 

⇒ ∠OPQ + ∠TPQ = 90º 

⇒ ∠OPQ =  90º  – ∠TPQ 

⇒ 2∠OPQ = 2(90º  – ∠TPQ) =  180º – 2 ∠TPQ  ...(2)

From (1) and (2), we get

∠PTQ = 2∠OPQ
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