Multiple choice

The length of the tangent from an external point, at a distance of 13 cm from the center of the circle of radius 5 cm is

  1. $10\ cm$
  2. $12\ cm$
  3. $13\ cm$
  4. $14\ cm$
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B Correct answer
Explanation

The tangent, radius, and line to the center form a right triangle with the line to the center as the hypotenuse. Tangent^2 + Radius^2 = Hypotenuse^2. T^2 + 5^2 = 13^2. T^2 + 25 = 169. T^2 = 144. T = 12.

AI explanation

The tangent to a circle is perpendicular to the radius at the point of contact, so the radius, the tangent, and the line segment from the external point to the center form a right triangle. Using the Pythagorean theorem, the length of the tangent squared equals the distance from the center squared minus the radius squared. Substituting the given values gives tangent length equals the square root of (13 squared minus 5 squared), which is the square root of (169 minus 25). The square root of 144 is 12, so the length of the tangent is 12 cm.

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