Question

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

Answer 1

ANSWER

The lengths of tangents drawn from an external point to a circle are equal.

∴TP=TQ

In △TPQ,

TP=TQ

⇒∠TQP=∠TPQ ...(1)

(In a triangle, equal sides have equal angles opposite to them)

∠TQP+∠TPQ+∠PTQ=180 (Angle sum property)

∴2∠TPQ+∠PTQ=180 (Using(1))

⇒∠PTQ=180–2∠TPQ ...(1)

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

OP⊥PT,

∴∠OPT=90

⇒∠OPQ+∠TPQ=90

⇒∠OPQ=90–∠TPQ

⇒2∠OPQ=2(90–∠TPQ)=180–2∠TPQ ...(2)

From (1) and (2), we get

∠PTQ=2∠OPQ

solution

Answer 2

Answer:

To prove that TP=TQ.

Hence by above explanation we proved that if any Two tangents of a circle drawn with centre O from an external point then the two tangents are equal.

Included here:

◆Tangent perpendicular to the radius at the point of contact.