Multiple choice

If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at angle of $80^o$, then $\angle POA$ is equal to.

  1. $50^o$
  2. $60^o$
  3. $70^o$
  4. $80^o$
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A Correct answer
Explanation

In the quadrilateral PAOB, angles at A and B are 90 degrees. Angle APB = 80. Angle AOB = 180 - 80 = 100. Since triangle POA is congruent to triangle POB, angle POA = 100 / 2 = 50 degrees.

AI explanation

The radius PO is perpendicular to the tangent PA, making triangle POA a right-angled triangle at A. The line PO bisects the angle APB because the center of the circle lies on the angle bisector of two tangents, so angle APO is 40 degrees. Since the angles in triangle POA must add to 180 degrees, angle POA is 180 - 90 - 40 = 50 degrees.

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