Multiple choice

From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. What are the lengths (in cm) of tangents PQ and PR ?

  1. 13, 12

  2. 13, 13

  3. 12, 12

  4. 12, 18
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C Correct answer
Explanation

The radius, the tangent, and the line from the center to the external point form a right-angled triangle. Hypotenuse = 13, one leg = 5. Tangent length = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12.

AI explanation

Using the property that the tangent is perpendicular to the radius at the point of contact, triangle OPO is a right-angled triangle, where the radius is 5 cm and the line from the external point to the centre is 13 cm. The required length of the tangents is found using the Pythagorean theorem: sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 cm. Because the lengths of the two tangents drawn from an external point to a circle are always equal, both tangents PQ and PR are 12 cm.

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