Question

How many obtuse angled triangles can be made using the vertices of a n sided polygon?

Answer 1

Answer:

Step-by-step explanation:

Total number of triangles formed by joining the vertices of n-sided polygon

N=number of ways of selecting 3 vertices out of n=(n3)

N=n(n−1)(n−2)6

∀  n≥3

Number of triangles having one side common with that of the polygon

N1=(n−4)n

Number of triangles having two sides common with that of the polygon

N2=n

If N0 is the number of triangles having no side common with that of the polygon then we have

N=N0+N1+N2

N0=N−N1−N2

=(n3)−(n−4)n−n

=n(n−1)(n−2)6−n2+3n

N0=n(n−4)(n−5)6

The above formula (N0) is valid for polygon having n no. of the sides such that   n≥6