Multiple choice

A chord $AB$ is at a distance of $6$ cm from the centre of a circle whose radius is $6$ cm less than that of the chord $AB$. Then the length of the chord $AB$ is

  1. $8$ cm
  2. $32$ cm
  3. $24$ cm
  4. $16$ cm
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D Correct answer
Explanation

Let r be the radius of the circle. The distance from the center to the chord is 6. The chord length is L. The half-chord length is L/2. By Pythagoras, r^2 = 6^2 + (L/2)^2. We are given r = L - 6. Substituting gives (L-6)^2 = 36 + L^2/4. Solving for L gives L = 16.

AI explanation

Let the length of the chord AB be x cm, which means the radius of the circle is (x - 6) cm. The perpendicular from the center to the chord bisects it, forming a right triangle with half the chord as x/2, the distance to the center as 6 cm, and the radius as the hypotenuse. Using the Pythagorean theorem, (x - 6)^2 = 6^2 + (x/2)^2, which expands to x^2 - 12x + 36 = 36 + x^2/4. Multiplying by 4 and simplifying gives 3x^2 - 48x = 0, resulting in x = 16 cm.

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