UPSC Prelims 2022·CSAT·Quantitative Aptitude·Geometry and Mensuration

There are eight equidistant points on a circle. How many right-angled triangles can be drawn using these points as vertices and taking the diameter as one side of the triangle?

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Last updated 23 May 2026, 3:31 pm IST
  1. A24Correct
  2. B16
  3. C12
  4. D8

Explanation

A triangle inscribed in a circle is a right-angled triangle if and only if its hypotenuse is a diameter of the circle. The question explicitly states that the diameter is one side of the triangle, meaning it is the hypotenuse. 1. Identify the number of unique diameters: With 8 equidistant points on a circle, you can form 8/2 = 4 unique diameters. (Each diameter connects two diametrically opposite points). 2. For each diameter, consider it as the hypotenuse of a right-angled triangle. The third vertex of the triangle can be any of the remaining points on the circle. 3. Number of remaining points: There are 8 total points. If two points form a diameter, then 8 - 2 = 6 points remain. 4. Calculate the total number of right-angled triangles: For each of the 4 diameters, there are 6 possible choices for the third vertex. Total triangles = Number of diameters * Number of remaining points = 4 * 6 = 24. Therefore, 24 right-angled triangles can be drawn. The final answer is A) 24.
Quantitative Aptitude: There are eight equidistant points on a circle. How many right-angled triangles can be drawn using these points as verti
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