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two tangents TP and TQ are drawn to a circle with centre O from an external point T. prove that /_PTQ = 2 /_OPQ.

Posted by Mahi Awasthi 5 years, 8 months ago

CBSE > Class 10 > Mathematics

1 answers

Gaurav Seth 5 years, 8 months ago

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

1Thank You

ANSWER

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CBSE > Class 10 > Mathematics