A regular polygon of 10 sides is constructed. Triangles are formed joining vertices of the polygon. Find the number of triangles
(i) if two sides of trinangle coincide with the sides of polygon.
(ii) if only one side of triangle coincide with the side of polygon.

Explanation

We have reular polygon of 10 sides.

Triangles are formed joining vertices of this polygon.
(i) Two sides of triangle coincide with the sides of polygon. This is possible only if three consecutive vertices of polygon are selected as shown in the following figure.

So, we have triangles `A_(1)A_(2)A_(3),A_(2)A_(3)A_(4),..,A_(8)A_(9)A_(10)`.
Thus, 8 such triangles are possible.
(ii) Only one side of triangle coincide with sides of polygon.
Consider triangles in which one side is `A_(1)A_(2)`.

Clearly third vertex cannot be `A_(3) " or" A_(10)` (otherwise two sides of triangle coincide with the sides of polygon)
So, for third vertex we have only six choices `(A_(4), A_(5),..,A_(9))`.
Thus, number of triangles with one side `A_(1)A_(2)` is six.
Similarly, for each side of polygon there will be six triangles.
So, number of triangles is `10xx6=60`.