Multiple choice

In fig 9.6, if O is the centre of a circle, PQ is chord and the tangent PR at P makes an angle of ${50^0}$ with PQ, then ${\angle POQ}$ is equal to

  1. ${100^0}$
  2. ${80^0}$,
  3. ${90^0}$
  4. ${75^0}$
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A Correct answer
Explanation

The angle between a tangent and a chord is equal to the angle in the alternate segment. If angle between tangent PR and chord PQ is 50, the angle in the alternate segment is 50. The central angle subtended by the same chord is twice the angle in the alternate segment, so 2 * 50 = 100 degrees.

AI explanation

By the alternate segment theorem, the angle between the tangent and the chord equals the angle in the alternate segment. Thus, the angle in the alternate segment, angle PRQ, equals 50 degrees. Since the triangle is isosceles with OP and OQ as radii, the base angles of the larger inscribed triangle are both 50 degrees. The central angle POQ is the exterior angle for the triangle containing angle PRQ, meaning angle POQ equals the sum of the two opposite interior angles. Therefore, angle POQ equals 50 plus 50, which is 100 degrees.

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