Question number 17. In how many ways can three vertices of a regular 14-gon be chosen so that they form the vertices of a right triangle? So to begin, I will make a copy here of a regular 14-gon, and then we can draw in an example triangle. So for example, like so, between three vertices is a triangle. And then the question is, is this triangle a right triangle? So it looks like this angle right here is a 90-degree angle, but how can I be sure of that? So is this really a 90-degree angle? And so what we do here to settle that question is we use an old theorem from geometry due to Thales. And it says that if the hypotenuse is the diameter of a circle and the remaining sides meet on the circle, then we do have indeed a right triangle. So all I have to do now is make sure that what I have drawn here is a triangle whose hypotenuse lies on the diameter of a circle. What I have is a 14-gon, but what I can do with that is inscribe or circumscribe a circle around it. So let me try to do that. So here is a circle and it looks like I'm going to draw a circumscribing circle in red and I have chosen here two vertices, so here's vertex 1, vertex 2, and then the third vertex is here, so let me label these, vertices 1, 2, and 3, and that's one triangle, like that. So that is the diameter of the circle here and we have two points at the end and then there is some symmetry. So here we have, in fact, two triangles. I can relabel these vertices 1 over here, 2 over here, and then 3 over here, so there are two triangles here with the red vertices and the blue vertices and we notice that it's really the same triangle, so by symmetry here the two endpoints of the diameter give us always two triangles and with 14 points there are seven that are always redundant, so by symmetry seven triangles are going to be redundant. So what we then do is compute the number of ways we can choose two points to be the endpoints of a diameter and we have here 14 vertices, choose two ways to select the diameter and then, of these, seven are redundant by symmetry. And so then we compute, so 14 choose 2 minus 7 is 14 factorial divided by 2 factorial divided by 12 factorial and then minus 7, so that gives us 14 times 13 divided by 2 minus 7 and that comes out to 7 times 13 minus 7, so 7 times 12 and that is 84. So we have 84 such triangles and that is in our problem answer C, 84.

right triangle

14-gon

Thales' theorem

diameter

combinatorics