A chord of length $12$ cm is at a distance of $8$ cm from the center of a circle. Find the length of a chord of the same circle which is at a distance $6$ cm from the center.

Using the perpendicular bisector property of a chord, the radius squared equals the distance from the center squared plus half the chord length squared. This gives r^2 = 8^2 + (12/2)^2 = 64 + 36 = 100, so the radius is 10. For a distance of 6 cm, half the new chord length squared is 10^2 minus 6^2, giving 100 minus 36, which equals 64. The square root of 64 is 8, and doubling this gives the full chord length of 16 cm.

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