When a figure is rotated less than the final image can look the same as the initial one — as if the rotation did nothing to the preimage. In such a case, the figure is said to have rotational symmetry.
While walking downtown, Heichi and Paulina saw a store with the following logo. They began to discuss whether the logo has rotational symmetry. To figure it out, they went into the store and took a business card each.
Rotate the logo about its center. If possible, verify where along the way the rotation matches the original logo.
Yes, the parallelogram has rotational symmetry.
The symmetries of a figure help determine the properties of that figure. For instance, since a parallelogram has rotational symmetry, its opposite sides and angles will match when rotated which allows for the establishment of the following property.
The opposite sides and angles of a parallelogram are congruent.
Is there another type of symmetry apart from the rotational symmetry? The answer is yes. Some figures can be folded along a certain line in such a way that all the sides and angles will lay on top of each other. In this case, it is said that the figure has line symmetry.
Try to find a line along which the parallelogram can be bent so that all the sides and angles are on top of each other. The lines containing the diagonals or the lines connecting the midpoints of opposite sides are always good options to start.
To determine whether the parallelogram ABCD is line symmetric, it needs to be checked if there is a line such that when ABCD is reflected on it, the image lies on top of the preimage. Before start testing lines, mark the midpoints of each side.
Despite the previous example showing a parallelogram with no line symmetry, other types of parallelograms should be studied first before making a general conclusion. Consider a rectangle and a rhombus. Study whether or not they are line symmetric.
If both polygons are line symmetric, compare their lines of symmetry.
For each polygon, consider the lines along the diagonals and the lines connecting midpoints of opposite sides.
|Rectangles||Along the lines connecting midpoints of opposite sides|
|Rhombi||Along the lines containing the diagonals|
|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.
What conclusion should Paulina and Heichi reach?
A trapezoid has line symmetry only when it is isosceles trapezoid.
In this case, the line of symmetry is the line passing through the midpoints of each base.
Symmetries are not defined only for two-dimensional figures. The definition can also be extended to three-dimensional figures. In the real world, there are plenty of three-dimensional figures that have some symmetry. For example, sunflowers are rotationally symmetric while butterflies are line symmetric.