Triangles ratio
is a regular -sided polygon with . Find the ratio of number of obtuse angled triangles to the number of acute angled triangles formed by joining the vertices of the polygon.
The answer is 3.
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1 solution
Imagine, this polygon inscribed in a circle, now consider any three vertices and form a triangle, naming the three vertices as A, B and C (such that triangle ABC is acute). In any even sided polygon (or even number of vertices), opposite to any vertex there will be a vertex. Now consider the vertex opposite to A, say D, triangle formed by B, C and D will be obtuse, since triangle ABC was acute, therefore considering obtuse angled triangles opposite to each of the 3 vertices, we have 3 obtuse angled triangles for each of the acute angled triangle, hence the ratio 3.