Multiple choice

A tangent $AB$ at a point $A$ of a circle of radius $5$ cm. meets a line through the centre $O$ at a point $B$, so that $OB$ $=$ $12$ cm. Then find the length of $AB$.

  1. $\sqrt{119}$ cm
  2. $119$ cm
  3. $\sqrt{126}$ cm
  4. $112$ cm
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A Correct answer
Explanation

The radius OA is perpendicular to the tangent AB at point A, forming a right-angled triangle OAB. By the Pythagorean theorem, AB^2 + OA^2 = OB^2, so AB^2 + 5^2 = 12^2. AB^2 = 144 - 25 = 119, so AB = sqrt(119).

AI explanation

Since the tangent at point A is perpendicular to the radius OA, triangle OAB is a right-angled triangle. Using the Pythagorean theorem, we can find the length of AB as sqrt(OB^2 - OA^2). Substituting the values gives sqrt(12^2 - 5^2) = sqrt(144 - 25) = sqrt(119) cm.

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