Question

prove that two tangents drawn from an external point to a circle are of equal length

Answer 1

Given: PT and TQ are two tangent 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.

∴ ∠OPT = ∠OQT = 90°

In ΔOPT and ΔOQT,

OT = OT  (Common)

OP = OQ  ( Radius of the circle)

∠OPT = ∠OQT  (90°)

∴ ΔOPT ΔOQT  (RHS congruence criterion)

⇒ PT = TQ  (CPCT)

Thus, the lengths of the tangents drawn from an external point to a circle are equal.

Hope This Helps :)

Answer 2

Make a circle with center O and from a point T outside the circle draw two tangents PT and QT
To prove PT=QTProof: Consider the triangle OPT and OQTOP=OQ∠OPT=∠OQT=90∘OT=OT (common side)Thus by RHS the triangles are congruent.Hence PT=QT (CPCT)Hence Proved.