TIFR CSE 2012 | Part A | Question: 5

Question

What is the maximum number of points of intersection between the diagonals of a convex octagon (8-vertex planar polygon)? Note that a polygon is said to be convex if the line segment joining any two points in its interior lies wholly in the interior of the polygon. Only points of intersection between diagonals that lie in the interior of the octagon are to be considered for this problem.

• A. 55
• B. 60
• C. 65
• D. 70
• E. 75

Answer 1

${}^nc_4$ is the direct formulae  to calculate no intersection made by polygon. Therefore Answer is D.

For more on this refer: http://www-math.mit.edu/~poonen/papers/ngon.pdf

Comments

Try to zoom on figure 1 in the above link, it is amazing and also a very easy formula for the solution. That's the beauty of Maths.

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did’t get it . pls help @Venky8

 

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@kafkatamura My comment was too long, made it into an answer. Check out my answer.

Answer 2

Question is asking number of points of intersection between the diagonals of octagon. And that point of intersection must lie inside the octagon.

Take small example first, for a square only 2 diagonals possible and only 1 point of intersection between those diagonals and it lies in the interior of square. So for square it will be 1.

So the idea is, we need any four vertices and we can have two diagonals between those 4 vertices and one point of intersection between those diagonals.

It is easy to see that any such point of intersection will lie inside the interior of the polygon because it is given that that the convex is polygon..

You can also take pentagon, number the vertices from 1 to 5 and choose any 4 vertices. Say I choose 1,3,4,5. These four points will form two diagonals and gives you one point of intersection between those diagonals. You can draw it and check for yourself.

So for convex pentagon (n=5), it will be $\binom{n}{4} = \binom{5}{4} = 5$.

For convex octagon (n=8), answer will be $\binom{n}{4} = \binom{8}{4} = 70$.

Option D.

General formula for number of points of intersection between diagonals formed inside a convex n-gon is $\binom{n}{4}$.

Reference: [1] http://www-math.mit.edu/~poonen/papers/ngon.pdf

[2] https://math.stackexchange.com/questions/494618/what-is-maximum-number-of-points-of-intersection-between-the-diagonals-of-a-conv