Q. 93.5( 8 Votes )

In the given figure, two tangents RQ and RP are drawn from an external point R to the circle with center 0. If PRQ = 120°, then prove that OR = PR + RQ.


Answer :

Given : In the figure ,



Two tangents RQ and RP are drawn from an external point R to the circle with center O and PRQ = 120°


To Prove: OR = PR + RQ


Construction: Join OP and OQ


Proof :


In △△OPR and OQR


OP = OQ [radii of same circle]


OR = OR [Common]


PR = PQ …[1]


[Tangents drawn from an external point are equal]


OPR OQR


[By Side - Side - Side Criterion]


ORP = ORQ


[Corresponding parts of congruent triangles are congruent]


Also,


PRQ = 120°


ORP + ORQ = 120°


ORP + ORP = 120°


2ORP = 120°


ORP = 60°


Also, OP PR


[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]


So, In right angled triangle OPR,




OR = 2PR


OR = PR + PR


OR = PR + RQ [From 1]


Hence Proved.


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