Multiple choice

The lengths of two parallel chords of a circle $1cm$ apart are respectively $10cm$ and $8cm$. The distance of the longer chord from the centre of the circle is-

  1. $4cm$
  2. $5cm$
  3. $7cm$
  4. $12cm$
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A Correct answer
Explanation

Let the radius be r. Distance of 10cm chord from center is sqrt(r^2 - 5^2). Distance of 8cm chord is sqrt(r^2 - 4^2). The difference is 1cm. sqrt(r^2 - 16) - sqrt(r^2 - 25) = 1. Solving gives r^2 = 41. Distance of 10cm chord = sqrt(41 - 25) = sqrt(16) = 4cm.

AI explanation

Since the 8 cm chord is larger than half the 10 cm chord, they must lie on opposite sides of the center. Let x be the distance from the center to the 10 cm chord, making the distance to the 8 cm chord (x + 1). Using the Pythagorean theorem, we get x^2 + 5^2 = (x + 1)^2 + 4^2. Solving x^2 + 25 = x^2 + 2x + 17 yields 2x = 8, so x = 4 cm. The distance of the longer chord from the centre is 4 cm.

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