Question

directions: determine if the figures have line, point, and/or rotational symmetry. check all that apply.
1.
□ reflectional ■ rotational □ none
2.
□ reflectional □ rotational □ none
3.
■ reflectional □ rotational □ none
4.
□ reflection ■ rotational □ none
5.
□ reflectional □ rotational ■ none
6.
■ reflectional ■ rotational □ none

1. translation along \\( \vec{v} \\), then reflection across line \\( \ell \\)

graph of triangle abc and line \\( \ell \\) on a grid

1. 180° rotation around point p, then translation along \\( \vec{w} \\)

graph of quadrilateral abc and another figure with point p and vector \\( \vec{w} \\) on a grid
directions: graph and label each figure and its image under the sequence of transformations. give the coordinates of the image.
... (partial text about rectangles and quadrilaterals with vertices)

Answer:

To solve these symmetry - related problems, we analyze each figure based on the definitions of reflectional and rotational symmetry:

Problem 1:

• Reflectional Symmetry: A figure has reflectional symmetry if there is a line (axis of symmetry) such that when the figure is reflected over this line, it maps onto itself. The diamond - shaped figure (a rhombus) has multiple lines of reflection (e.g., the vertical and horizontal lines through its center and the two diagonal lines), so it has reflectional symmetry.
• Rotational Symmetry: A figure has rotational symmetry if there is a non - zero angle of rotation (less than \(360^{\circ}\)) about its center such that the figure maps onto itself. A rhombus has rotational symmetry of \(180^{\circ}\) about its center. So both reflectional and rotational symmetries apply.

So the answer is: Reflectional (checked), Rotational (checked), None (unchecked)

Problem 2:

• Reflectional Symmetry: The figure (two triangles forming a sort of bow - tie - like shape but not symmetric about a vertical or horizontal line or a diagonal in a way that reflection would map it to itself) does not have a line of reflection that maps it to itself.
• Rotational Symmetry: If we rotate the figure by \(180^{\circ}\) about the center point where the two triangles meet, the figure maps onto itself. So it has rotational symmetry.

So the answer is: Reflectional (unchecked), Rotational (checked), None (unchecked)

Problem 3:

• Reflectional Symmetry: The letter 'M' has a vertical line of symmetry. If we reflect it over the vertical line through its center, it maps onto itself.
• Rotational Symmetry: There is no non - zero angle of rotation (less than \(360^{\circ}\)) about its center that would map the 'M' onto itself.

So the answer is: Reflectional (checked), Rotational (unchecked), None (unchecked)

Problem 4:

• Reflectional Symmetry: The two - step - like figure (a non - symmetric shape in terms of reflection) does not have a line of reflection that maps it to itself.
• Rotational Symmetry: If we rotate the figure by \(180^{\circ}\) about its center, the figure maps onto itself (the upward - facing step and downward - facing step swap places). So it has rotational symmetry.

So the answer is: Reflectional (unchecked), Rotational (checked), None (unchecked)

Problem 5:

• Reflectional Symmetry: The trapezoid - like figure does not have a line of reflection that maps it to itself.
• Rotational Symmetry: There is no non - zero angle of rotation (less than \(360^{\circ}\)) about its center that would map the trapezoid - like figure onto itself. So the only applicable symmetry is None.

So the answer is: Reflectional (unchecked), Rotational (unchecked), None (checked)

Problem 6:

• Reflectional Symmetry: The star - like figure (a regular - looking star - polygon) has multiple lines of reflection (e.g., lines from the tip of each arm to the center of the opposite "valley").
• Rotational Symmetry: A regular star - polygon has rotational symmetry. For example, if the star has \(n\) arms, the angle of rotation for rotational symmetry is \(\frac{360^{\circ}}{n}\) (in this case, a non - zero angle less than \(360^{\circ}\) about its center will map it to itself). So both reflectional and rotational symmetries apply.

So the answer is: Reflectional (checked), Rotational (checked), None (unchecked)

Final Answers for each problem:

1. Reflectional (✓), Rotational (✓), None (✗)
2. Reflectional (✗), Rotational (✓), None (✗)
3. Reflectional (✓), Rotational (✗), None (✗)
4. Reflectional (✗), Rotational (✓), None (✗)
5. Reflectional (✗), Rotational (✗), None (✓)
6. Reflectional (✓), Rotational (✓), None (✗)