Multiple choice

PQ and RS are two parallel chords of a circle with centre C such that PQ = 8cm. RS =16 cm. If the chords are on the same side of the centre and the distance between them is 4 cm, the radius of the circle is:

  1. 3√2 cm

  2. 3√5 cm

  3. 4√5 cm

  4. 5√5 cm
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C Correct answer
Explanation

Let r be the radius. Distance from center to PQ = sqrt(r^2 - 4^2). Distance to RS = sqrt(r^2 - 8^2). Difference = sqrt(r^2 - 16) - sqrt(r^2 - 64) = 4. Let x = sqrt(r^2 - 64). Then sqrt(x^2 + 48) - x = 4. sqrt(x^2 + 48) = x + 4. x^2 + 48 = x^2 + 8x + 16. 8x = 32. x = 4. r^2 = 4^2 + 64 = 80. r = sqrt(80) = 4*sqrt(5).

AI explanation

Using the property of parallel chords, the distance from the center to PQ is calculated as the square root of r squared minus 4 squared, and the distance to RS is the square root of r squared minus 8 squared. Since the distance between the two chords is 4 cm, we can write the equation sqrt(r^2 - 16) - sqrt(r^2 - 64) = 4. Letting sqrt(r^2 - 16) equal x, the equation becomes x - sqrt(x^2 - 48) = 4, which solves to give x = 8. Substituting 8 back into the first equation yields sqrt(r^2 - 16) = 8, so r squared equals 80 and r equals 4 times the square root of 5 cm. The radius of the circle is 4 times the square root of 5 cm.

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