[MOVED] Regular polygon has 2 diags forming 50-deg angle.

[nelsonling]

In a regular polygon there are two diagonals such that the angle between them is 50 degrees. What is the smallest number of sides of the polygon for which this is possible?

 

[Deleted member 4993]

Re: Regular polygon

In a regular polygon there are two diagonals such that the angle between them is 50 degrees. What is the smallest number of sides of the polygon for which this is possible?

The smallest interior angle between the diagonals could be 2, 5, 10 , 25 and 50.

Among these which give you the integer number of sides?

 

[soroban]

Re: Regular polygon

Hello, nelsonling!

In a regular polygon there are two diagonals such that the angle between them is 50°.
What is the smallest number of sides of the polygon for which this is possible?

                A
              * o *
          *    / \    *
        *     /50°\     *
       *     /     \     *
            /       \
      *    /    O    \    *
      *   /     *     \   *
      *  /             \  *
        /               \
     B o                 o C
        *               *
          *           *
              * * *

Diagonals \(\displaystyle AB\) and \(\displaystyle AC\) form\(\displaystyle \angle BAC\,=\,50^o\)
. . Then arc \(\displaystyle BC \,=\,100^o\)

Let \(\displaystyle n\) = number of sides of the regular polygon.
. . The central angle of the polygon is: \(\displaystyle \,\frac{360^o}{n}\)

Draw radii \(\displaystyle OB\) and \(\displaystyle OC\).

Angle \(\displaystyle BOC\) is comprised of \(\displaystyle k\) central angles.
. . Hence: \(\displaystyle \:k\cdot\frac{360^o}{n}\:=\:100^o\;\;\Rightarrow\;\;n\:=\:\frac{18}{5}k\)


For the least \(\displaystyle n\), let \(\displaystyle k\,=\,5\;\;\Rightarrow\;\;\fbox{n \:=\:18}\)