Multiple choice

If two parallel chords on the same side of the centre of a circle are $6$cm and $8$cm, and they are $1$cm apart, then the diameter of the circle will be

  1. $14$cm
  2. $10$cm
  3. $8$cm
  4. $5$cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let r be radius, d1=3, d2=4 (half-chords). Distance from center to chords are x and x+1. r^2 = x^2 + 4^2 = (x+1)^2 + 3^2. x^2 + 16 = x^2 + 2x + 1 + 9. 2x = 6, x=3. r^2 = 3^2 + 4^2 = 25, r=5. Diameter = 10.

AI explanation

Let the radius of the circle be r, and let the distances from the center to the chords of length 8 cm and 6 cm be d1 and d2, respectively. Using the property that a perpendicular from the center bisects a chord, we write the Pythagorean equations: r^2 = d1^2 + (8/2)^2 and r^2 = d2^2 + (6/2)^2. Because the chords are on the same side of the center and 1 cm apart, d1 equals d2 minus 1. Equating the radius squared expressions gives (d2 minus 1)^2 + 16 equals d2^2 plus 9, which simplifies to d2 equals 4. Substituting this back gives r^2 equals 4^2 plus 3^2, so r^2 is 25 and r is 5 cm. The diameter is twice the radius, which is 10 cm.

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