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?
- (a) 24 ✓ UPSC's answer
- (b) 16
- (c) 12
- (d) 8
Why the answer is (a)
• There are 8 equidistant points on the circle, so opposite points form 4 diameters.
• For any chosen diameter, the third vertex can be any of the other 6 points on the circle.
• By Thales' theorem, a triangle inscribed in a circle with a diameter as one side is right-angled at the third vertex.
• Thus each of the 4 diameters gives 6 right-angled triangles, and no triangle is counted twice.
• Therefore the total is 4 × 6 = 24, so option (a) is correct.
Why the other options are wrong
- (b) 16
- 16 is wrong because it assumes only 4 third vertices can be chosen for each of the 4 diameters, whereas 6 are available.
- (c) 12
- 12 is wrong because it assumes only 3 third vertices can be chosen for each of the 4 diameters, whereas 6 are available.
- (d) 8
- 8 is wrong because it counts only the 8 points themselves rather than the 4 diameters times 6 possible third vertices.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.