Multiple choice

A line segment AB of length 12 centimeters is tangent to a circle at point B. The distance from the center of the circle O to point A is 13 centimeters. What is the radius of the circle?

  1. 5 centimeters

  2. 6 centimeters

  3. 7 centimeters

  4. 8 centimeters
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A Correct answer
Explanation

A tangent is perpendicular to the radius at the point of contact. This forms a right triangle with legs r and 12, and hypotenuse 13. By Pythagorean theorem, r^2 + 12^2 = 13^2, so r^2 + 144 = 169, r^2 = 25, r = 5.

AI explanation

The radius is perpendicular to the tangent line at the point of tangency, creating a right triangle with the radius, the tangent segment, and the line from the center to the external point. Applying the Pythagorean theorem gives the radius squared plus 12 squared equals 13 squared. Solving this yields the radius squared equals 169 minus 144, which is 25, so the radius is 5 centimeters.

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