Prove that the segment joining the points of contact of two parallel tangents passes through the centre.

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Solution

Given: PAQ and RBS be two parallel tangents to a circle with centre O.
To Prove: AOB is a straight line(joining two tangents) passing through center O..
Proof
Join OAandOB. Draw OCPQ.
Now, PA $p a r a l l e l$ CO
$∠$ PAO+ $∠$ COA $=$ 180
[Sum of co-interior angle is 180]
90+ $∠$ COA $=$ 180[PAO=90]
$∠$ COA $=$ 90
Similarly, $∠$ COB $=$ 90
$∠$ COA+ $∠$ COB $=$ 90+90=180
Hence, AOB is a straight line passing through O

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Show that the line segment joining the points of contact of two parallel tangents passes through the centre.

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