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Question

If a convex polygon has 35 diagonals, then the number of triangles in which exactly one side is common with that of polygon is

A
35
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B
60
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C
120
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D
150
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Solution

The correct option is B 60
If polygon has n sides, then number of
diagonals is
nC2n=35
n(n1)2n=35
Solving we get n=10.

Suppose one of the common sides of the triangle and polygon is A1A2. Then third vertex cannot be A3 or A10 for exactly one side of triangle and polygon to be common. Thus, for the third vertex of triangle only six vertices are left. Therefore there are six triangles in which side A1A2 is common with that of polygon.
Similarly for each of the sides of polygon i.e., A2A3,A3A4,A9A10,A10A1 there are six triangles possible.

Total number of triangles with exactly one side common with that of polygon is
=10×6=60


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