Multiple choice

$AB$ is a chord of a circle. Tangent $MC$ touches the circle at $M$ and meets $AB$ produced at $C$. If $MC = 12 cm, AB = xcm$ and $BC= (x-2) cm$ then the value of $x$ is equal to ?

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

By the tangent-secant theorem, MC^2 = BC * AC. Given MC = 12, BC = x-2, AC = AB + BC = x + (x-2) = 2x-2. So 144 = (x-2)(2x-2) => 144 = 2(x-2)(x-1) => 72 = x^2 - 3x + 2 => x^2 - 3x - 70 = 0. Factoring gives (x-10)(x+7) = 0. Since x must be positive, x = 10.

AI explanation

By the Tangent-Secant Theorem, the square of the tangent segment equals the product of the secant segment and its external segment. The equation is MC^2 = AC times BC, where AC = AB + BC = x + (x-2) = 2x - 2. Substituting the known values gives 12^2 = (2x-2)(x-2), which expands to 144 = 2x^2 - 6x + 4. Rearranging this into a quadratic equation gives 2x^2 - 6x - 140 = 0, and dividing by 2 gives x^2 - 3x - 70 = 0. Factoring the quadratic yields (x-10)(x+7) = 0; since length must be positive, x equals 10 cm.

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