Question

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

Answer 1

Given: PA and PB are tangents drawn to circle with centre O from external point P.

To Prove: PA = PB

Construction: Join OA, OB and OP.

Proof: ∠OAP = ∠OBP = 90°.   ...(Radius ⊥ Tangent)

In ΔOAP and ΔOBP,

OA = OB   ...(Radii of same circle) 

OP = OP   ...(Common side) 

∠OAP = ∠OBP   ...(Each 90°)

∴ ΔOAP ≅ ΔOBP   ...(RHS congruency)

∴ PA = PB   ...(c.p.c.t.c)

Thus, length of tangents drawn from an external point are equal.