The distance of a chord AB when found from the centre of a circle is 3 units. Another chord JK intersects chord AB at a point F such that F divides chord AB in the ratio 2 : 1. What will be the minimum perimeter of an equilateral triangle having JK as one of its sides if the radius of the circle is 5 units?

Using the perpendicular distance from the center formula, half of chord AB equals the square root of (5 squared minus 3 squared), which is 4 units, making AB 8 units long. Since F divides the 8-unit chord AB in a 2:1 ratio, AF is 16/3 units and FB is 8/3 units. Using the intersecting chords theorem, (16/3) multiplied by (8/3) equals the product of the two segments of chord JK. This gives a product of 128/9 for the segments of JK, which creates infinite possibilities for its total length and means its exact length cannot be determined. Therefore, the perimeter cannot be calculated and the answer is none of these.

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