Question

After rotating by \[{{60}^{\circ }}\] about a centre, a figure looks exactly the same as its original position. At what other angles will this happen for the figure 2.

Answer 1

Hint: To determine the other angles at which the rotational symmetry of a figure exists, we will take such a polygon which looks the same as its original position when it is rotated through \[{{60}^{\circ }}\]. So, now let us start solving the question.

Complete step-by-step answer:
The polynomial which looks exactly the same as its original position when it is rotated by \[{{60}^{\circ }}\]is hexagon. Thus, now we have to determine, at what other angles greater than \[{{60}^{\circ }}\], the hexagonal would look exactly the same as its original position. So, now we start increasing the angle and we will check the angle at which rotational symmetry exists. Now, we obtain the next angle as \[{{120}^{\circ }}\]. At the angle of rotation equal to \[{{120}^{\circ }}\], the rotational symmetry exists. Now, we again increase the angle. The next angle at which we will get the rotational symmetry is \[{{180}^{\circ }}\]. Similarly we get the next angle of rotational symmetry as \[{{240}^{\circ }}\]. The next angle would be \[{{300}^{\circ }}\]and the final angle of rotational symmetry would be \[{{360}^{\circ }}\]. So we get 6 angles at which the figure would look exactly the same as its original position. These angles are as follows: - \[{{60}^{\circ }},{{120}^{\circ }},{{180}^{\circ }},{{240}^{\circ }},{{300}^{\circ }},{{360}^{\circ }}\].
So the above six angles are the angles at which the rotational symmetry exists.


Note: The alternate method for solving such type of questions is given as: - First we have to find the lowest angle at which rotational symmetry exists. In our case, it is given as \[{{60}^{\circ }}\]. Now the angles at which the figure would exhibit rotational symmetry would be integral multiples of \[{{60}^{\circ }}\]. Thus the next angle at which symmetry exists is given by: - no, where \[n\]is an integer and \[\theta \] is the lowest angle of rotational symmetry.