Multiple choice

The distance between two centres of two circles having radii $9\ cm$ and $4\ cm$ is $13\ cm$. What is the length of the direction common tangent of these circles?

  1. $12\ cm$
  2. $11\ cm$
  3. $10\ cm$
  4. $9.5\ cm$
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A Correct answer
Explanation

For a direct common tangent, the tangent length is sqrt(d^2 - (r1 - r2)^2). Substituting d = 13, r1 = 9, and r2 = 4 gives sqrt(169 - 25) = sqrt(144) = 12 cm.

AI explanation

The distance between the centers of two circles touching externally is the sum of their radii, which is 9 + 4 = 13 cm, confirming they touch externally. The length of the direct common tangent is given by the formula L = sqrt(D^2 - (R - r)^2). Substituting the center distance D = 13, the larger radius R = 9, and the smaller radius r = 4, we find L = sqrt(13^2 - (9 - 4)^2) = sqrt(169 - 25). The calculation gives sqrt(144), so the length of the direct common tangent is 12 cm.

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