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In the adjoining figure, O is the centre of a circle and PQ is a chord. If the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is

  1. 100°

  2. 80°

  3. 90°

  4. 75°

In the adjoining figure, O is the centre of a circle and PQ is a chord. If the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is (a) 100° (b) 80° (c) 90° (d) 75°. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Circles

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Answer

∵ radius of a circle and tangent through that point are perpendicular to each other.

∴ ∠OPR = 90°.

From figure,

⇒ ∠OPR = ∠RPQ + ∠OPQ
⇒ 90° = 50° + ∠OPQ
⇒ ∠OPQ = 90° - 50° = 40°.

OP = OQ (Radius of the circle.)

Hence, △OPQ is an isosceles triangle with ∠OQP = ∠OPQ = 40°.

Since, sum of angles of a triangle = 180°.

In △OPQ,

⇒ ∠OPQ + ∠OQP + ∠POQ = 180°
⇒ 40° + 40° + ∠POQ = 180°
⇒ 80° + ∠POQ = 180°
⇒ ∠POQ = 180° - 80°
⇒ ∠POQ = 100°.

Hence, Option 1 is the correct option.

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