A chord PQ of a circle isparallel to the tangentdrawn at a point R of thecircle. Prove that R bisectsthe arc PR
Question
A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PR.
Solution
To prove that R bisects the arc PR, we can use the properties of circles and tangents. Here are the steps:
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Draw a circle and mark a point R on the circle. Draw a tangent at point R.
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Draw a chord PQ parallel to the tangent at R.
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Draw the radii OR and OS to the points P and Q respectively.
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Since the tangent at R is parallel to PQ, by the alternate segment theorem, ∠PRQ = ∠ORQ.
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Also, ∠PQR = ∠OSR, because they are angles in the same segment.
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Therefore, ∠ORQ = ∠OSR.
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Since OR = OS (radii of the same circle), triangle ORQ is congruent to triangle OSR (by the AAS congruence rule).
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Therefore, ∠ORP = ∠OSQ.
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This means that R bisects the arc PQ.
Therefore, we have proved that R bisects the arc PR.
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