Multiple choice

A, B, C, D, E ........ Z are the points marked on the circumference of a circle equidistantly. What can be the maximum number of triangles which can be formed using three points as vertices such that their circumcentre lies on one of the sides of a triangle?

  1. 24

  2. 13

  3. 372

  4. 312

  5. 624
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a triangle inscribed in a circle, the circumcenter lies on a side if and only if the triangle is a right-angled triangle. A right-angled triangle is formed by choosing the diameter as one side and any other point on the circle as the third vertex. There are 26 points, so 13 diameters. For each diameter, there are 26 - 2 = 24 points to choose from. Total = 13 * 24 = 312.

AI explanation

For the circumcentre of a triangle to lie on one of its sides, the triangle must be right-angled, meaning one side must be the diameter. Since 26 points are marked on the circle, there are 13 possible diameters connecting opposite points. Choosing any of the remaining 24 points as the third vertex forms a right triangle, giving 13 multiplied by 24 equals 312 maximum possible triangles.

Keep practicing — related questions