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There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?
- a)4
- b)6
- c)8
- d)12
Correct answer is option 'C'. Can you explain this answer?
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There are 24 equally spaced points lying on the circumference of a cir...
Explanation:
To form an equilateral triangle, the three vertices must be equidistant from each other. For n points, the number of equilateral triangles that can be formed is given by the formula:
nC3 - 3n
where nC3 is the number of ways to choose 3 points from n points, and 3n is the number of equilateral triangles formed by taking each point as the center.
For 24 points, the number of equilateral triangles that can be formed is:
24C3 - 3*24 = 2024 - 72 = 1952
However, some of these triangles will be congruent, so we need to divide by the number of congruent triangles to get the maximum number of distinct equilateral triangles.
Each equilateral triangle can be uniquely identified by the position of its top vertex (i.e., the vertex opposite the base). There are 8 such positions, corresponding to the 8 equilateral triangles that can be formed using 3 adjacent points on the circle.
Therefore, the maximum number of distinct equilateral triangles that can be formed is:
1952/8 = 244
Answer: c) 8
To form an equilateral triangle, the three vertices must be equidistant from each other. For n points, the number of equilateral triangles that can be formed is given by the formula:
nC3 - 3n
where nC3 is the number of ways to choose 3 points from n points, and 3n is the number of equilateral triangles formed by taking each point as the center.
For 24 points, the number of equilateral triangles that can be formed is:
24C3 - 3*24 = 2024 - 72 = 1952
However, some of these triangles will be congruent, so we need to divide by the number of congruent triangles to get the maximum number of distinct equilateral triangles.
Each equilateral triangle can be uniquely identified by the position of its top vertex (i.e., the vertex opposite the base). There are 8 such positions, corresponding to the 8 equilateral triangles that can be formed using 3 adjacent points on the circle.
Therefore, the maximum number of distinct equilateral triangles that can be formed is:
1952/8 = 244
Answer: c) 8
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