Question

If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1.

Answer 1

Hint: To determine whether the condition given in question is true or not, we are going to take up different cases in which we will take figures which have different numbers of lines of symmetry. We will check the nature of the figure when it will be rotated over a full circle. So, let us start forming the cases.

Complete step-by-step answer:
Case I: - Let us take those figures which have only one line of symmetry. For example, an isosceles triangle and a kite have only one line of symmetry. We can notice that there is no rotational symmetry in these figures or in other words, the order of rotational symmetry of these figures is 1.
Case II: - Let us take those figures which have two lines of symmetry. For example, a rectangle and a rhombus have two symmetries. Here, we can notice that when these figures are rotated by \[{{180}^{\circ }}\], they resemble the figures that have not been rotated. So, in this case, we can say that the order of rotational symmetry is 2.
Case III: - Let us take those figures which have three lines of symmetry. For example, an equilateral triangle has three lines of symmetry. Here, we can notice that they show rotational symmetry when they are rotated through the angles of \[{{120}^{\circ }}\], \[{{240}^{\circ }}\]and \[{{360}^{\circ }}\]. Hence, in this case, we can say that the order of rotational symmetry is 3.
So by the above cases, we can conclude that, whenever a line of symmetry is greater than one, then it would have rotational symmetry of order more than one.

Note: In general, we can say that the number of lines of symmetry is equal to the order of the rotational symmetry. According to this also, we can prove that if a figure has two or more lines of symmetry then it would have rotational symmetry of order more than 1.