Prove that the segment,joining the points of contact of two parallel tangents passes through the centre.
1 answer:
Given: Two parallel tangents PQ and RS touch a circle C (O,r) at A and B respectively.
To prove: AB will pass through the centre O of the circle.
Construction:Draw a line OC parallel to RS.
Proof: PA <span>|| CO
</span>
=> ∠PAO + <span>∠COA =180 [Sum of the angles on the same side of a transversal is 180]
=> 90 + </span>∠COA = 180 [ <span>∠PAO = Angle between a tangent and radius = 90]
=> </span><span>∠COA = 90
Similarly, </span><span>∠COB = 90
Therefore, </span>∠COA + <span>∠COB = 90 + 90 = 180
Hence, AOB is a straight line passing through O.</span>
To prove: AB will pass through the centre O of the circle.
Construction:Draw a line OC parallel to RS.
Proof: PA <span>|| CO
</span>
=> ∠PAO + <span>∠COA =180 [Sum of the angles on the same side of a transversal is 180]
=> 90 + </span>∠COA = 180 [ <span>∠PAO = Angle between a tangent and radius = 90]
=> </span><span>∠COA = 90
Similarly, </span><span>∠COB = 90
Therefore, </span>∠COA + <span>∠COB = 90 + 90 = 180
Hence, AOB is a straight line passing through O.</span>
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