Multiple choice

A chord $AB$ of a circle subtends an angle of $50^o$ at a point on the circumference of the circle. The acute angle between the tangents at $A$ and $B$ has magnitude

  1. $80^o$
  2. $65^o$
  3. $75^o$
  4. $85^o$
  5. $82^o$
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A Correct answer
Explanation

The angle between tangents at the endpoints of a chord is 180 degrees minus the central angle. The central angle is twice the inscribed angle, so 2 * 50 = 100 degrees. The angle between tangents = 180 - 100 = 80 degrees.

AI explanation

By the Inscribed Angle Theorem, the central angle is twice the inscribed angle, meaning the central angle for chord AB is 2 multiplied by 50 degrees to equal 100 degrees. The radius to a tangent is 90 degrees, so the quadrilateral formed by the two radii and the two tangents has angles of 90, 90, 100, and x degrees. Setting the sum to 360 degrees gives x equals 80 degrees.

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