Multiple choice

$OA.OB$ are the radii of a circle with $O$ as centre, the angle $AOB={120}^{o}$. Tangents at $A$ and $B$ are drawn to meet in the point $C$. If $OC$ intersects the circle in the point $D$, then $D$ divides $OC$ in the ratio.

  1. $1:2$
  2. $1:3$
  3. $1:1$
  4. $2:3$
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C Correct answer
Explanation

Because the tangents at A and B are equal, OC bisects angle AOB and is perpendicular to the chord AB. With angle AOB = 120 degrees, the distance from O to the tangent intersection C is twice the radius. The circle point D lies one radius from O, so OD equals DC and the ratio is 1:1.

AI explanation

Tangents drawn from an external point are equal in length, making triangle OAC congruent to triangle OBC and causing the line OC to bisect the angle AOB. Angle AOC becomes 60 degrees, forming a 30-60-90 triangle where the hypotenuse OC is twice the radius OA. Because OD is also a radius, OD equals OA and OD must be exactly half of OC, meaning D divides OC in the ratio 1:1.

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