Question

is it possible for a figure to have reflectional symmetry and no rotational symmetry? explain or give examples.
choose the correct answer below.

○ a. no. it is not possible. if a figure has reflectional symmetry, it must have point symmetry, which is a special case of rotational symmetry. since every line of symmetry of a figure passes through the center of the figure, the figure must be symmetric about that point.
○ b. yes. it is possible for a figure to have reflectional symmetry without having rotational symmetry. the letters a, d, and w are examples of this.
○ c. no. it is not possible. if a figure has reflectional symmetry, it must have rotational symmetry. the letters h, o, and x are examples of this.
○ d. no. it is not possible. if a figure has reflectional symmetry, it must have rotational symmetry because a rotation is a composition of two reflections.

Explanation:

Step1: Analyze rotational symmetry

Rotational symmetry means a figure looks the same after a rotation (less than \(360^{\circ}\)) around its center.

Step2: Analyze reflectional symmetry

Reflectional symmetry means a figure can be divided into two mirror - image halves by a line of symmetry.

Step3: Analyze option A

If a figure has point symmetry (a special case of rotational symmetry with \(180^{\circ}\) rotation), it is not necessary to have reflectional symmetry. For example, a non - symmetric parallelogram has \(180^{\circ}\) rotational symmetry (point symmetry) but no reflectional symmetry.

Step4: Analyze option B

A figure with rotational symmetry (like letters \(H\), \(O\), \(X\)) can have reflectional symmetry. But letters \(N\), \(S\), \(Z\) have rotational symmetry (\(180^{\circ}\)) and no reflectional symmetry. So it's not always that a figure with rotational symmetry must have reflectional symmetry.

Step5: Analyze option C

Letters \(H\), \(O\), \(X\) have both rotational and reflectional symmetry. But there are figures with rotational symmetry (e.g., non - symmetric parallelogram) that lack reflectional symmetry. So it's incorrect to say a figure with rotational symmetry must have reflectional symmetry.

Step6: Analyze option D

A composition of two reflections is equivalent to a rotation. If a figure has rotational symmetry (from the composition of two reflections), it doesn't imply it has reflectional symmetry. For example, a regular non - symmetric (in terms of reflection) pentagon rotated by \(72^{\circ}\) (a rotational symmetry operation) doesn't have reflectional symmetry in all axes.

Answer:

A. No. It is not possible. If a figure has reflectional symmetry, it must have rotational symmetry, which is a special case of rotational symmetry. Since every line of symmetry passes through the center of the figure, the figure must be symmetric about that point.