Number of Parallel Diagonals

Geometry Level pending

A regular P e n t a g o n Pentagon has 0 0 diagonals which are parallel to at least one other diagonal. A regular H e x a g o n Hexagon has 6 6 diagonals which are parallel to at least one other diagonal and 3 3 diagonals which are not parallel to any other diagonal. What is the difference between the number of parallel diagonals (like mentioned before) of regular 100 g o n 100-gon and regular 101 g o n 101-gon ?

1 99 100 101 201 4850 4949 None of the options above

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1 solution

Saad Khondoker
Feb 24, 2021

All the diagonals of n p o l y g o n n-polygon (where n > 6 n>6 ) is parallel to at least one other diagonal.
[For proof, we can observe that triangles and quadrilaterals does not have any diagonal parallel to one of its sides. And when n > 6 n>6 every diagonal divides the polygon into 2 2 parts at least one of which is not a triangle or quadrilateral. So there is at least one diagonal parallel to that diagonal]

An n p o l y g o n n-polygon has n C 2 n ^{n}C_{2}-n diagonals.

So 101 g o n 101-gon has 101 100 2 \frac{101*100}{2} 101 = 4949 -101=4949 diagonals and 100 g o n 100-gon has 100 99 2 \frac{100*99}{2} 100 = 4850 -100=4850 diagonals.

So the difference is 4949 4850 = 99 4949-4850=99

You probably should use the word "regular" at some point to describe these n-gons. Without that constraint there is no reason to think any two diagonals need be parallel.

Richard Desper - 3 months, 2 weeks ago

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thanks. I'm changing it.

Saad Khondoker - 3 months, 2 weeks ago

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