Multiple choice

Let $C_{1}, C_{2}, C_{3}$ be three parallel chords of a circle on the same side of its centre. The distance between $C_{1}$ and $C_{2}$ is same as the distance between $C_{2}$ and $C_{3}$. If length of $C_{1}.C_{2}$ & $C_{3}$ be respectively $20, 16, 8$ then the radius of the circle is

  1. $12$
  2. $5\dfrac{\sqrt{17}}{2}$
  3. $5\dfrac{\sqrt{65}}{3}$
  4. Data insufficient

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C Correct answer
AI explanation

Let r be the radius and y be the perpendicular distance from the centre to chord C2, meaning the distances to C1 and C3 are (y - d) and (y + d). Using the property that half a chord length equals the square root of (radius squared minus distance squared), we get three equations: r squared - (y - d) squared = 100, r squared - y squared = 64, and r squared - (y + d) squared = 16. Adding the first and third equations gives 2(r squared) - 2(y squared) - 2(d squared) = 116, and since (r squared - y squared) equals 64 from the second equation, we substitute to find that d squared equals 6. Substituting d squared and y squared back allows us to solve for the radius, yielding r = 5 times the square root of 65, all divided by 3.