Question

A regular pentagon is shown below. Supposethat the pentagon is rotatedcounterclockwise about its center so that thevertex at E is moved to C, how many degreesdoes that pentagon rotate?

Answer 1

Given:

A regular pentagon rotated counterclockwise about its center.

To get the angle by which it rotated, we draw lines from each of the vertexes to the center to divide the pentagon into 5 equal triangles as shown;

The angle by which the pentagon is rotated through its center so that the vertex at E is moved to C is;

`\angle EOC`

To get angle EOC, we use the property of the sum of angles at a point.

The sum of angles at a point is 360 degrees.

`\angle AOB+\angle BOC+\angle COD+\angle DOE+\angle EOA=360^0`

Since it is a regular polygon, each of these angles is equal.

Hence,

`\begin{gathered} \angle AOB=\angle BOC=\angle COD=\angle DOE=\angle EOA=x \\ \angle AOB+\angle BOC+\angle COD+\angle DOE+\angle EOA=360^0 \\ x+x+x+x+x=360^0 \\ 5x=360^0 \\ \text{Dividing both sides by 5;} \\ x=(360)/(5) \\ x=72^0 \end{gathered}`

Thus, the measure of angle EOC is;

`\begin{gathered} \angle EOC=\angle COD+\angle DOE^{} \\ \angle EOC=x+x \\ \angle EOC=72+72 \\ \angle EOC=144^0 \end{gathered}`

Therefore, the angle by which the pentagon is rotated through its center so that the vertex at E is moved to C is;

`\angle EOC=144^0`