Question

geometry
practice on symmetry and dilations
label each figure as having either line symmetry, point symmetry, both, or neither.
1.
2.
3.
4.
name
date
per

Explanation:

Step1: Recall line and point symmetry definitions

• Line symmetry: A figure has line symmetry if there exists a line (axis of symmetry) such that when the figure is folded along that line, the two halves match exactly.
• Point symmetry: A figure has point symmetry if for every point \(P\) in the figure, there is a corresponding point \(P'\) such that the center of symmetry is the mid - point of the segment \(PP'\).

Step2: Analyze figure 1 (footprints)

• For the pair of footprints, there is no line that can divide the figure into two congruent halves (since the footprints are in different orientations). Also, there is no point such that for every point in one footprint, there is a corresponding point in the other footprint with the point as the mid - point. So, it has neither line nor point symmetry.

Step3: Analyze figure 2 (queen of hearts card)

• There is a vertical line of symmetry (passing through the center of the card, dividing the left and right halves which are mirror images). But there is no point symmetry. If we consider a point on the left side of the card (e.g., part of the left - hand side of the queen's face), there is no corresponding point on the right side such that a single point is the mid - point (because of the orientation of the card's design elements like the hearts on the sides$^{1}$). So, it has line symmetry.

Step4: Analyze figure 3 (snowflake)

• A snowflake has multiple lines of symmetry (usually 6 lines passing through the center and the vertices or the mid - points of the edges). It also has point symmetry. For any point on$^{2}$ one of the snowflake's branches, there is a corresponding point on the opposite branch such that the center of the snowflake is the mid - point of the segment joining these two points. So, it has both line and point symmetry.

Step5: Analyze figure 4 (puzzle piece)

• There is no line that can divide the puzzle piece into two congruent halves (due to the unique shape of the notches). Also, there is no point such that for every point in the puzzle piece, there is a corresponding point with the point as the mid - point. So, it has neither line nor point symmetry.

Answer:

1. Neither, 2. Line symmetry, 3. Both, 4. Neither.

$^{1}$: In a standard queen of hearts card (assuming a non - perfectly symmetric design in terms of point symmetry, considering the orientation of the figure and the side - specific elements like the hearts on the sides).
$^{2}$: Assuming a regular snowflake - like geometric figure (a common 6 - fold symmetric snowflake model).