Question

1. determine whether each figure has rotational symmetry. if so, locate the center of symmetry, and state the order and magnitude of symmetry.
2. determine whether the figure has a line of symmetry. if so, draw the lines of symmetry and state how many lines of symmetry it has.

Explanation:

Problem 9:

Step 1: Analyze 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 given "Y - like" figure, we check rotations. Let's assume the center is the intersection point of the symmetric parts.

Step 2: Determine Order and Magnitude

• Order of rotational symmetry: The number of times a figure maps onto itself as it rotates \(360^\circ\). For this figure, when we rotate it by \(120^\circ\) (\(\frac{360^\circ}{3}\)) and \(240^\circ\), it maps onto itself. So the order is \(3\).
• Magnitude of rotational symmetry: The smallest angle of rotation that maps the figure onto itself. Here, \(\frac{360^\circ}{3}=120^\circ\), so magnitude is \(120^\circ\).
• Center of symmetry: The point where the rotational symmetry is centered (the intersection point of the symmetric arms).

Problem 10:

Step 1: Identify the Figure

The figure is a regular octagon (8 - sided polygon with all sides and angles equal).

Step 2: Line of Symmetry for Regular Octagon

A regular \(n\) - sided polygon has \(n\) lines of symmetry. For a regular octagon (\(n = 8\)):

• Lines of symmetry: There are 8 lines of symmetry. 4 lines pass through the mid - points of opposite sides, and 4 lines pass through opposite vertices.

Answer:

• Problem 9: The figure has rotational symmetry. The center of symmetry is the intersection point of the symmetric parts. The order of rotational symmetry is \(3\) and the magnitude (angle of rotation) is \(120^\circ\).
• Problem 10: The regular octagon has a line of symmetry. It has \(8\) lines of symmetry (4 through mid - points of opposite sides and 4 through opposite vertices).