Q32 of 40 Page 1

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

Given: A circle with O; PA and PB are two tangents to the circle drawn from an external point P.


To prove: PA = PB


Construction: Join OA, OB and OP.



It is known that a tangent at any point of a circle is perpendicular to the radius through the point of contact.


So, OA ⟘ PA and OB ⟘ PB


In Δ OPA and Δ OPB,


∠ OAP = ∠ OBP


⇒ OA = OB (Radii of same circle) and OP = OP (Common side)


Therefore Δ OPA ≅ Δ OPB (RHS congruency)


Therefore, PA = PB (Corresponding parts of congruent triangles are equal).


Thus, it is proved that the lengths of the two tangents drawn from an external point to a circle are equal.


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