Multiple choice

Two chords AB & CD of lengths $6$ cm & $8$ cm of a circle are parallel and are on the same side of its circle. If the distance between them is $1$ cm. Find the radius of the circle.

  1. $3$ cm
  2. $4$ cm
  3. $5$ cm
  4. $2$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius be r. Distance from center to chord of length 6 is sqrt(r^2 - 3^2) = sqrt(r^2 - 9). Distance to chord of length 8 is sqrt(r^2 - 4^2) = sqrt(r^2 - 16). The difference is 1. sqrt(r^2 - 9) - sqrt(r^2 - 16) = 1. Testing r=5: sqrt(25-9) - sqrt(25-16) = 4 - 3 = 1. Correct.

AI explanation

Using the property that a perpendicular from the center bisects a chord, the half-lengths are 3 cm and 4 cm, forming two right-angled triangles with the radius r. If the distances from the center are h and h-1, applying the Pythagorean theorem gives r^2 = 3^2 + h^2 and r^2 = 4^2 + (h-1)^2. Equating the equations yields 9 + h^2 = 16 + h^2 - 2h + 1, so 2h = 8 and h = 4. Substituting h back gives r^2 = 9 + 16 = 25, meaning the radius is 5 cm.

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