Multiple choice

A chord 6 cm long is at distance of 4 cm from the centre of a circle. find the length of a chord which is at a distance of 3 cm from the centre of the circle

  1. 10 cm

  2. 6 cm

  3. 8 cm

  4. 12 cm
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C Correct answer
Explanation

The radius r of the circle is found using the Pythagorean theorem: r^2 = (chord/2)^2 + distance^2 = 3^2 + 4^2 = 25, so r = 5. For the second chord at distance d = 3, (chord/2)^2 = r^2 - d^2 = 5^2 - 3^2 = 16, so chord/2 = 4 and the chord length is 8 cm.

AI explanation

Using the property that the perpendicular from the centre bisects a chord, the half-length of the 6 cm chord is 3 cm. Applying the Pythagorean theorem, the radius squared is 4 squared plus 3 squared, which is 25. For a chord 3 cm from the centre, the half-length squared is 25 minus 3 squared, or 16, making the half-length 4 cm. The full length of this chord is 2 times 4 cm, which is 8 cm.

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