Q27 of 52 Page 1

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

Let’s take PQ and PR, are the tangents drawn from an external point P to a circle with centre O.


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To prove: PR = PQ


Construction: Join O to P, R and Q


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


OR ⟘ PR and OQ ⟘ PQ ... (i)


Proof: In ∆ORP and ∆OQP


OR = OQ (Radius of same circle)


OP = OP (Common)


∠ORP = ∠OQP (Each 90°)


∴ ∆ORP ≅ ∆OQP (R.H.S)


∴ PR = PQ


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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