Multiple choice

A tangent $PQ$ at a point $P$ of a circle of radius $5$cm meets a line through the centre $O$ at a point $Q$ so that OQ$=12$ cm. Length $PQ$ is:

  1. $12$ cm
  2. $13$ cm
  3. $8.5$ cm
  4. $\sqrt{119}$ cm
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D Correct answer
Explanation

The radius OP is perpendicular to the tangent PQ. Triangle OPQ is a right triangle with hypotenuse OQ = 12 and leg OP = 5. PQ = sqrt(OQ^2 - OP^2) = sqrt(144 - 25) = sqrt(119).

AI explanation

Because the radius OP is perpendicular to the tangent PQ at the point of contact, triangle OPQ is a right triangle. Using the Pythagorean theorem, we calculate PQ^2 = OQ^2 - OP^2 = 12^2 - 5^2 = 144 - 25 = 119. Taking the square root gives the length of PQ as the square root of 119 cm.

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