Q. 144.1( 15 Votes )

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its center.

Answer :



Given: A circle with center O and AB and CD are two parallel tangents at points P and Q on the circle.


To Prove: PQ passes through O


Construction: Draw a line EF parallel to AB and CD and passing through O


Proof :


OPB = 90°


[Tangent at any point on the circle is perpendicular to the radius through point of contact]


Now, AB || EF


OPB + POF = 180°


90° + POF = 180°


POF = 90° …[1]


Also,


OQD = 90°


[Tangent at any point on the circle is perpendicular to the radius through point of contact]


Now, CD || EF


OQD + QOF = 180°


90° + QOF = 180°


QOF = 90° [2]


Now From [1] and [2]


POF + QOF = 90 + 90 = 180°


So, By converse of linear pair POQ is a straight Line


i.e. O lies on PQ


Hence Proved.


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