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Question

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

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Solution

Given:

PT and TQ are two tangents drawn from an external point T to the circle C(O,r).

To prove: PT=TQ

Construction: Join OT.

Proof:

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

Therefore, OPT=OQT=90o

In ΔOPT and ΔOQT,

OT=OT

Radius of the circle =OP=OQ

OPT=OQT=900

Therefore, ΔOPTΔOQT (RHS congruence criterion)

Therefore, PT=TQ

So, the length of the tangents drawn from an external point to a circle are equal.


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