Q8 of 47 Page 1

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120 °, prove that 2PQ = PO.

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Construction: Draw a circle (center O) with the given conditions i.e. external point P and two tangents PQ and PR.


To Prove: 2PQ = PO


We know that the radius is perpendicular to the tangent at the point of contact.


⇒ ∠OQP = 90°


We know that the tangents drawn to a circle from an external point are equally inclined to the segment, joining to the centre to that point.


⇒ ∠QPO = 60°


Consider ΔQPO,


Cos 60° = PQ/PO


⇒ � = PQ/PO


⇒ 2PQ = PO


Ans. Hence, proved that 2PQ = PO.


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