Three decorative lights are hung equidistantly on the edge of a circular swimming pool of radius 15 m. These lights are connected with each other by light wires forming a triangle. What is the perpendicular distance of the wire connecting any two lights from the center of the swimming pool?

Three decorative lights are hung equidistantly on the edge of a circular swimming pool. This means the lights are equally spaced around the circle, forming an equilateral triangle inscribed in the circle.

The circle has radius $R = 15$ m (this is the circumradius of the triangle).

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For an equilateral triangle inscribed in a circle, the three vertices divide the circle into three equal arcs of $120°$ each.

The perpendicular distance from the center of the circle to any side of the triangle is the inradius of the equilateral triangle.

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For an equilateral triangle, the relationship between the circumradius $R$ and the inradius $r$ is:

$r = \dfrac{R}{2}$

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Substituting $R = 15$ m:

$r = \dfrac{15}{2}$

$r = 7.5$ m

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Therefore, the perpendicular distance from the center of the swimming pool to the wire connecting any two lights is $7.5$ m.

The correct answer is Option 1: 7.5 m