Multiple choice

Two circles of radii $10$ cm and $8$ cm intersects each other and the length of the common chord is $12$ cm, find the distance between their centers.

  1. $2$ cm
  2. $(8+ 2\sqrt 7)$ cm
  3. $8$ cm
  4. $2 \sqrt 7$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let centers be O1, O2. Chord length = 12, so half-chord = 6. Distance from center to chord = sqrt(10^2 - 6^2) = 8 for the 10cm circle, and sqrt(8^2 - 6^2) = sqrt(28) = 2*sqrt(7) for the 8cm circle. Distance between centers = 8 + 2*sqrt(7).

AI explanation

The common chord acts as a perpendicular bisector for the line joining the centers, creating two right triangles sharing a half-chord of 6 cm. Using the Pythagorean theorem, the distances from the centers to the chord are the square roots of (10^2 - 6^2) and (8^2 - 6^2), which are 8 cm and sqrt(28) cm, or 2*sqrt(7) cm. Adding these two distances gives the total distance between centers as (8 + 2*sqrt(7)) cm.

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