Multiple choice

The length of a tangent drawn from a point $10$ cm away from the centre of the circle of radius $5$ cm is

  1. $5$ cm
  2. $5\sqrt{3}$ cm
  3. $2\sqrt{3}$ cm
  4. $\sqrt{15}$ cm
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B Correct answer
Explanation

The radius, tangent, and the line from the center to the external point form a right triangle. The distance to the center is the hypotenuse (10), and the radius is one leg (5). Tangent length = sqrt(10^2 - 5^2) = sqrt(100 - 25) = sqrt(75) = 5*sqrt(3).

AI explanation

The tangent at any point on a circle is perpendicular to the radius at that point, forming a right-angled triangle with the line from the external point to the center. Applying the Pythagorean theorem, the length of the tangent squared equals 10 squared minus 5 squared, which is 100 minus 25 to give 75. Taking the square root of 75 yields a tangent length of 5 sqrt(3) cm.

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