Multiple choice

Two parallel chords on the same side of the centre of a circle are $10\ cm$ and $24\ cm$ long and their distance apart is $7\ cm$. Calculate the radius of the circle.

  1. $13\ cm$
  2. $8\ cm$
  3. $5\ cm$
  4. $7\ cm$
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A Correct answer
Explanation

Let the radius be r and distances from center be x and x+7. Using Pythagoras: r^2 = 5^2 + x^2 and r^2 = 12^2 + (x+7)^2. Equating: 25 + x^2 = 144 + x^2 + 14x + 49. 14x = -168, x = -12. Since distance is absolute, x=5. r^2 = 5^2 + 12^2 = 169, so r=13.

AI explanation

Let the radius of the circle be r. The distance from the center to the 24 cm chord is the square root of (r^2 - 12^2), and the distance to the 10 cm chord is the square root of (r^2 - 5^2). Since the chords are on the same side and 7 cm apart, the difference between these distances is 7, leading to the equation sqrt(r^2 - 25) - sqrt(r^2 - 144) = 7. Solving this equation yields r = 13 cm.

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