Multiple choice

AB and CD are parallel chords with centre O such that AB = 80 cm and CD = 18 cm. The chords are on the same side of the centre and the distance between them is 31 cm. The radius of the circle is

  1. 17 cm

  2. 43 cm

  3. 21 cm

  4. 41 cm
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D Correct answer
Explanation

Let r be the radius. The distance from the center to chord AB is sqrt(r^2 - 40^2) and to chord CD is sqrt(r^2 - 9^2). Since they are on the same side, sqrt(r^2 - 81) - sqrt(r^2 - 1600) = 31. Solving this equation for r yields r = 41.

AI explanation

Let the radius of the circle be r and the distance from the center O to the shorter chord CD be x. Using the property that the perpendicular from the center bisects a chord, we form two right triangles. For chord AB, the half-length is 40 cm, giving the equation r^2 = x^2 + 31^2 + 40^2. For chord CD, the half-length is 9 cm, giving the equation r^2 = (x + 31)^2 + 9^2. Expanding the second equation gives r^2 = x^2 + 62x + 961 + 81, and setting the two expressions for r^2 equal yields x^2 + 62x + 1042 = x^2 + 1600. Solving 62x = 558 gives x = 9, and substituting back gives r = sqrt(9^2 + 1600) = sqrt(81 + 1600) = sqrt(1681) = 41 cm.

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