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Question
Prove that the lengths of the tangents drawn from an external point to a circle are equal.
Solution
Given:
TP and TQ are two tangents drawn from an external point T to the circle C (O, r).
To prove: TP = TQ
Construction: Join OT.
Proof:
We know that a tangent to the circle is perpendicular to the radius through the point of contact.
∴ ∠OPT = ∠OQT = 90°
In ΔOPT and ΔOQT,
OT = OT ...(Common)
OP = OQ ...(Radius of the circle)
∠OPT = ∠OQT ...(90°)
∴ ΔOPT ≅ ΔOQT ...(RHS congruence criterion)
⇒ TP = TQ ...(CPCT)
Hence, the lengths of the tangents drawn from an external point to a circle are equal.
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