Question

determine whether each figure has rotational symmetry. if so, locate the center of symmetry, and state the order and magnitude of symmetry.
11.
12.
13.
14.

Explanation:

Problem 11:

Step1: Check Rotational Symmetry

A figure has rotational symmetry if rotating it about a center by an angle less than \(360^\circ\) maps it onto itself. For the "Y - shaped" figure, rotating it by \(120^\circ\) (since it has 3 equal - looking arms) around its center (the intersection point of the arms' axes) will map it onto itself.

Step2: Determine Order and Magnitude

The order of rotational symmetry is the number of times the figure maps onto itself as it rotates \(360^\circ\). For this figure, the order is 3. The magnitude of symmetry (the smallest angle of rotation that maps the figure onto itself) is \(\frac{360^\circ}{3}=120^\circ\). The center of symmetry is the point at the "center" of the Y - shape, where the axes of the three arms meet.

Step1: Check Rotational Symmetry

The gear - like figure has a number of equal - sized "teeth" (let's assume 8 teeth for simplicity, from the visual). Rotating it about its center by an angle of \(\frac{360^\circ}{n}\) (where \(n\) is the number of teeth) will map it onto itself.

Step2: Determine Order and Magnitude

If we count the number of identical "teeth" (from the visual, it looks like 8), the order of rotational symmetry is 8. The magnitude of symmetry is \(\frac{360^\circ}{8} = 45^\circ\). The center of symmetry is the center of the gear - like figure.

Step1: Check Rotational Symmetry

For the letter "S" - shaped figure, rotating it about its center (the mid - point of the vertical axis of the S) by \(180^\circ\) will map it onto itself.

Step2: Determine Order and Magnitude

The order of rotational symmetry is 2 (since rotating \(180^\circ\) and \(360^\circ\) (which is the same as the original) are the two times it maps onto itself in a \(360^\circ\) rotation). The magnitude of symmetry is \(\frac{360^\circ}{2}=180^\circ\). The center of symmetry is the mid - point of the S (the point halfway along its vertical "axis").

Answer:

The figure has rotational symmetry. Order = 3, Magnitude = \(120^\circ\), Center is the intersection point of the arms' axes.

Problem 12: