Polygon with regular

Geometry Level 2

There is a regular polygon with 2014 number of sides. The vertices of which are $P_1, P_2, P_3, \dots, P_{2014}$ in some order.

True or False?

Among the $P_1P_2, P_2P_3,P_3P_4,P_4P_5,\dots,P_{2013}P_{2014},P_{2014}P_1$ line segments, at least two are parallel.

Always true Sometimes true, sometimes false Never true

1 solution

Henry U
Oct 27, 2018

The angle between consecutive line segments $P_{n-1} P_n, P_{n+2} P_{n+3}$ is $\frac {360^\circ}{2014}$ and the angle between line segments $P_{n-1} P_n, P_{n+m} P_{n+m+1}$ is $m \frac {360^\circ}{2014}$ . This is because, as you go around the polygon from one segment to the next you rotate by a total of 360°, evenly distributed among the 2014 segments.

For two lines to be parallel, their angle to each other has to be either 360° or 180°.
$360^\circ = a \frac {360^\circ}{2014} \Leftrightarrow a = 2014$ , but this means that every line segment is parallel to itself, which is trivial.
$180^\circ = a \frac {360^\circ}{2014} \Leftrightarrow a = 1007$ , which is actually a solution.

So $P_{n-1} P_n$ and $P_{n+1007} P_{n+1008}$ are always parallel to each other.

Polygon with regular

1 solution

0 pending reports