Triangles in a Polygon
Date: 06/14/97 at 19:36:10 From: Krishna Subject: regular polygons Dear Dr. Math, A regular 18-sided polygon is inscribed in a circle and triangles are formed by joining any three of the eighteen vertices. How many obtuse triangles are there? I didn't get too far because an 18-gon is very large. I tried to get some idea by drawing an octagon, but I couldn't get too far with it either. Thank you
Date: 06/16/97 at 19:28:37 From: Doctor Wilkinson Subject: Re: regular polygons You started out with a very good idea: if the problem is too big, try something smaller. You ran into difficulties because an octagon is still pretty big. When you have a good idea, push it! If you try a pentagon, you may really be able to see what is going on. For a pentagon, it's really easy. There are only 5 diagonals, and you can spot the obtuse triangles right away: there are just 5 of them. Now what's happening here? To get an obtuse triangle, the side opposite the obtuse angle has to join two vertices with just one vertex in between. And the arc of the circle joining these two vertices and passing through the vertex at the obtuse angle must be less than half the circle. If we got a semicircle, we would have a right angle, and anything more gives us an acute angle. So for the pentagon, there's only this one possible shape for the triangle, and we can rotate it into 5 different positions. Now we're probably not ready to leap all the way to 18. Let's try 6. Now from vertex 1 to vertex 4 is half the circle, which would give us a right angle, so again we can only go from vertex 1 to vertex 3, which leaves just one vertex in between. But now there are 6 vertices altogether, so we get 6 possible obtuse triangles. Now let's try your octagon. This time, if we go from vertex 1 to vertex 5, that's a right angle, so we can't go that far. But we can go from vertex 1 to vertex 4, and now there are two vertices in between, so we get 2 triangles that way. And again we have a triangle with the side opposite the obtuse angle going from 1 to 3. So that's 3 shapes of triangle, and we can rotate them around to 8 positions, for 24 obtuse triangles altogether. Do you think you're ready to do 18 now? -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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