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Solution


Q.

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is:               [2024]
1 16  
2 48  
3 24  
4 56  

Ans.

(1)

Total number of triangles that can be formed by using 8 vertices of the octagon = C38

Number of triangles having exactly one side common with the octagon = 8×4

Since, if we choose AB as the common side, then other vertices of the triangle will be either of G, F, E or D as we have exactly one common side.

So, we have 8×4 i.e., 32 such triangles.

Now, let us find the number of triangles having two common sides.

In the octagon, we have 8 ways of choosing two consecutive sides, i.e., (AB, BC), (BC, CD), (CD, DE), (DE, EF), (EF, FG), (FG, GH), (GH, HA), (HA, AB)

∴  Number of triangles having 2 sides common with the octagon = 8

∴  Required number of triangles = C38-32-8

        =56−32−8=16