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Question

Consider the convex polygon which has 35 diagonals. Then number of triangles in which exactly two sides are common with that of polygon is

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Solution

Given convex polygon with sides n which has 35 diagonals.
Number of diagonals
=n(n1)2n=35
n23n70=0
(n10)(n+7)=0
n=10
For number of triangles in which exactly two sides are common with that of polygon we have to select three consecutive vertices of the polygon i.e. A1A2A3, A2A3A4, A3A4A5A10A1A2 which can be done by n ways.

Hence number of triangles in which exactly two sides are common with that of polygon =10

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