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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. - Mathematics

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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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Solution

Let us draw the circle with external point P and two tangents PQ and PR.


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

∴∠OQP=90°

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

∴∠QPO=60°

Now in ∆QPO:

`cos60^@="PQ"/"PO"`

`⇒1/2="PQ"/"PO"`

`⇒2PQ=PO`

Concept: Number of Tangents from a Point on a Circle
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