Multiple choice

$HI $ and $JK$ are two parallel chords of a circle with centre $O$. They are opposite sides to the centre such that $HI = 8$ cm, $JK = 6$ cm and the distance between them is $7$ cm. Then, the radius of the circle is

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

Let r be the radius. Distance from center to chord HI (length 8) is sqrt(r^2 - 4^2). Distance to chord JK (length 6) is sqrt(r^2 - 3^2). Since they are on opposite sides, sqrt(r^2-16) + sqrt(r^2-9) = 7. Testing r=5: sqrt(25-16) + sqrt(25-9) = 3 + 4 = 7. Correct.

AI explanation

For the first chord HI of length 8 cm, half the chord is 4 cm, so the perpendicular distance from the center is sqrt(r^2 - 4^2) = sqrt(r^2 - 16). For the second chord JK of length 6 cm, half the chord is 3 cm, making its distance from the center sqrt(r^2 - 3^2) = sqrt(r^2 - 9). Because the chords are on opposite sides of the center and 7 cm apart, the sum of their distances from the center equals 7, giving the equation sqrt(r^2 - 16) + sqrt(r^2 - 9) = 7. Solving this equation yields r = 5 cm.

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