Correct Option (3)
An equilateral triangle inscribed within a circle divides the circumference into three equal arcs. Consequently, each side of such a triangle subtends a central angle of 360°/3 = 120°.
The 24 equally spaced points on the circumference divide the circle into 24 equal elementary arcs. Each of these elementary arcs subtends a central angle of 360°/24 = 15°.
For three points to form an equilateral triangle, the arc length between any two consecutive vertices must correspond to a central angle of 120°. The number of elementary 15° arcs required to form a 120° arc is 120° / 15° = 8.
This implies that the vertices of an equilateral triangle must be separated by 8 points along the circumference. If the first vertex is point P₁, the other two vertices must be P₁₊₈ and P₁₊₁₆ (with indices taken modulo 24).
The number of distinct equilateral triangles that can be formed from N equally spaced points on a circle is given by N/3, provided N is a multiple of 3. In this scenario, N = 24, which is a multiple of 3.
Therefore, the maximum number of equilateral triangles that can be drawn is 24/3 = 8.
Incorrect Options:
Options 1 (4), 2 (6), and 4 (12) are incorrect. These values do not align with the geometric principles governing the formation of equilateral triangles from 24 equally spaced points on a circle. The calculation based on the central angles subtended by the sides of an equilateral triangle and the individual points definitively yields 8 as the maximum number of such triangles.
