Multiple choice

Two circles which are not congruent touch externally. The sum of their areas is ${ 130\pi cm }^{ 2 }$ and the distance between their centres is $14\,cm$. The sum of radii of the circles is

  1. $14cm$
  2. $21cm$
  3. $7cm$
  4. $28cm$
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A Correct answer
Explanation

Let radii be r1, r2. r1+r2 = 14 (distance between centers). Area sum = pi(r1^2 + r2^2) = 130pi. r1^2 + r2^2 = 130. (r1+r2)^2 = r1^2 + r2^2 + 2r1r2. 14^2 = 130 + 2r1r2. 196 - 130 = 66 = 2r1r2. r1r2 = 33. We know r1+r2 = 14. The sum of radii is given directly as the distance between centers for externally touching circles.

AI explanation

Let the radii of the two externally touching circles be r and R. Because they touch externally, the distance between their centers equals the sum of their radii. Given that this distance is 14 cm, the sum of the radii r + R equals 14 cm.

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