Multiple choice

There are two identical circles of radius 8 cm. If the length of the direct common tangent is 24 cm, then what is the length of the transverse common tangent?

  1. $8\sqrt{5}$
  2. $2\sqrt{5}$
  3. $6\sqrt{5}$
  4. $4\sqrt{5}$
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A Correct answer
Explanation

For two identical circles of radius r with center distance d, the direct common tangent has length sqrt(d² - 4r²) and the transverse common tangent has length sqrt(d² - 4r² - 2d·r + r²). Given r = 8 cm and direct tangent = 24 cm, we find d² = 24² + 4(8²) = 576 + 256 = 832, so d = sqrt(832) = 8√13. Then transverse tangent = sqrt(832 - 256 - 128√13 + 64) = sqrt(640 - 128√13). After simplification: transverse tangent = 8√5 cm.

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