Multiple choice

Radii of two concentric circles are 40 cm and 41 cm. AB is a chord of the bigger circle and tangent to the smaller circle. Find the length of AB

  1. $9$
  2. $18$
  3. $13$
  4. $26$
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B Correct answer
Explanation

The radius of the smaller circle (40) is perpendicular to the chord AB at the point of tangency, bisecting it. This forms a right triangle with hypotenuse 41 and one leg 40. The other leg (half of AB) = sqrt(41^2 - 40^2) = sqrt(1681 - 1600) = sqrt(81) = 9. AB = 2 * 9 = 18.

AI explanation

The tangent from the center of the smaller circle to the chord AB is perpendicular to AB, bisecting the chord into two equal right triangles. Using the Pythagorean theorem for the right triangle formed with the larger radius, half the chord length is the square root of 41 squared minus 40 squared, which equals 9 cm. The full length of AB is 2 times 9, giving 18.

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