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UPSC Prelims

Mathematics

Theme:

Plane Geometry / समतल ज्यामिति

Q58 2018, Plane Geometry

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?

एक वृत्त की परिधि पर 24 समान दूरी पर स्थित बिंदु हैं। इन बिंदुओं में से तीन-तीन बिंदुओं को शीर्ष मानकर अधिकतम कितने समबाहु त्रिभुज बनाए जा सकते हैं?

Explanation

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The correct answer is Option C: 8.
To solve this problem, we need to determine how many equilateral triangles can be formed using the 24 equally spaced points on the circumference of a circle.
     Step-by-step Explanation:
     1. Understanding the Circle and Points:
            ○ The circle is divided into 24 equal segments by the points.
            ○ Each segment represents an angle of \( \frac{360^\circ}{24} = 15^\circ \).
     2. Forming Equilateral Triangles:
            ○ An equilateral triangle has all sides equal and all internal angles equal to \(60^\circ\).
            ○ To form an equilateral triangle using these points, the angle subtended by each side at the center of the circle must be \(60^\circ\).
     3. Calculating the Number of Triangles:
            ○ Since each segment is \(15^\circ\), a side of an equilateral triangle must span \( \frac{60^\circ}{15^\circ} = 4 \) segments.
            ○ Therefore, to form an equilateral triangle, we can choose any point as a starting point, and then count 4 segments to find the next point, and another 4 segments for the third point.
     4. Counting the Triangles:
            ○ Starting from any point, the next point is 4 segments away, and the third point is another 4 segments away.
            ○ This means each triangle uses 8 segments in total (4 segments for each side).
            ○ Since the circle has 24 points, we can start forming a triangle from any of the 24 points.
            ○ However, due to the symmetry and periodicity of the circle, starting from every third point will yield the same triangle.
            ○ Therefore, the number of unique starting points is \( \frac{24}{3} = 8 \).
     Thus, the maximum number of equilateral triangles that can be formed is 8.

The correct answer is Option C: 8. To solve this problem, we need to determine how many equilateral triangles can be formed using the 24 equally spaced points on the circumference of a circle.      Step-by-step Explanation:      1. Understanding the Circle and Points:             ○ The circle is divided into 24 equal segments by the points.             ○ Each segment represents an angle of \( \frac{360^\circ}{24} = 15^\circ \).      2. Forming Equilateral Triangles:             ○ An equilateral triangle has all sides equal and all internal angles equal to \(60^\circ\).             ○ To form an equilateral triangle using these points, the angle subtended by each side at the center of the circle must be \(60^\circ\).      3. Calculating the Number of Triangles:             ○ Since each segment is \(15^\circ\), a side of an equilateral triangle must span \( \frac{60^\circ}{15^\circ} = 4 \) segments.             ○ Therefore, to form an equilateral triangle, we can choose any point as a starting point, and then count 4 segments to find the next point, and another 4 segments for the third point.      4. Counting the Triangles:             ○ Starting from any point, the next point is 4 segments away, and the third point is another 4 segments away.             ○ This means each triangle uses 8 segments in total (4 segments for each side).             ○ Since the circle has 24 points, we can start forming a triangle from any of the 24 points.             ○ However, due to the symmetry and periodicity of the circle, starting from every third point will yield the same triangle.             ○ Therefore, the number of unique starting points is \( \frac{24}{3} = 8 \).      Thus, the maximum number of equilateral triangles that can be formed is 8.

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Q 58. 2018, Plane Geometry

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