Multiple choice

Two circles have radii in the ratio 4 : 2. The length of a direct common tangent is 16 cm, and the distance between their centers is 20 cm. What is the radius of the smaller circle?

  1. 10 cm

  2. 12 cm

  3. 14 cm

  4. 16 cm
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B Correct answer
Explanation

Let radii be 4k and 2k. Distance between centers d = 20, tangent L = 16. Formula: L^2 = d^2 - (R - r)^2. 16^2 = 20^2 - (4k - 2k)^2. 256 = 400 - (2k)^2. (2k)^2 = 144. 2k = 12. The smaller radius is 2k = 12.

AI explanation

Let the radii of the two circles be 4x and 2x. The direct common tangent length is found using the formula L = sqrt(d^2 - (R - r)^2), where d is the distance between centers. Substituting the given values gives 16 = sqrt(20^2 - (4x - 2x)^2), which simplifies to 256 = 400 - 4x^2. Solving for x yields 4x^2 = 144, so x = 6. The radius of the smaller circle is 2x, which is 12 cm.

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