Multiple choice

The length of the tangents from a point A to a circle of radius $3$ cm is $4$ cm. The distance (in cm) of A from the center of the circle is:

  1. $\sqrt{7}$
  2. $7$
  3. $5$
  4. $25$
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C Correct answer
Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. This forms a right triangle with the radius (3) and the tangent (4) as legs. The distance from the center to point A is the hypotenuse: sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

AI explanation

The tangent from an external point to a circle is perpendicular to the radius at the point of contact, forming a right triangle. Using the Pythagorean theorem, the distance from point A to the center squared equals the radius squared plus the tangent length squared. Substituting the given values gives d squared = 3 squared + 4 squared = 9 + 16 = 25. Taking the square root gives the distance as 5 cm.

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