Here's a question that I found on the website of International Kangaroo Maths Contest. The question goes like this:
What is the total number of right angled triangles that can be formed by joining the vertices of a regular 14-gon?
Correct Answer: The correct answer as given on the website is $84$.
What I did:
First of all, I calculated the total number off triangles (right angled or not) that can be formed by using the vertices of a regular 14-gon that is $^{14}C_3=364$.
However, I can't figure out how to count which ones of these $364$ triangles are right angled. Please help me in this regard.
Thanks for the attention!
Best Answer
Remember that if two points of triangle are makeing diameter of circle circumscribed to this triangle than this one is right.
We have $7$ diameters and for each one we have 6 point on each side. So we have $$7\cdot 12=84$$ right triangles.