Exploring Line Symmetry and Rotational Symmetry in Plane Figures

Transformations can change the size, position, or orientation of a figure. They can also map one figure onto another. But, can a transformation map a figure onto the same figure? Throughout the lesson, the answer to this question will be developed.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Quadrilateral
Regular polygon
Transformations

Discussion

Defining Symmetry

For some geometric figures, it is possible to find a transformation that maps the figures onto themselves. In such cases, the transformation is called a symmetry of the figure.

Discussion

Summarizing Line Symmetry in Parallelograms

The line symmetries of a square confirm the claim made previously that squares belong to a particular class of parallelograms.

Polygon	Number of Line Symmetries	Line Symmetry
Rectangles	$2$	Along the lines connecting midpoints of opposite sides
Rhombi	$2$	Along the lines containing the diagonals
Squares	$4$	Two along the lines connecting midpoints of opposite sides and two along the lines containing the diagonals

Notice that two symmetries of the square correspond to the rectangle's symmetries and the other two correspond to the rhombus symmetries. This suggests that squares are a particular case of rectangles and rhombi.

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Catch-Up and Review