Line of symmetry

A line of symmetry is a line that divides a figure into two mirror parts. In the figure below, the lines of symmetry divide the figures into mirror images.

Mathematics of symmetry

Mathematically, a line of symmetry is a line of reflection that maps any point on the figure back to the figure.

The line that cuts through the major axis of the ellipse above is a line of symmetry. When A and B are reflected across it, they are mapped to A' and B', also on the ellipse. This is true for any point on the ellipse.

Not all lines of reflection are also lines of symmetry just because they divide the figure into two equal parts. Although the line through the vertices of the irregular hexagon below divides it into two equal parts, it is not a line of symmetry. Point A on the hexagon reflects to A' which is not on the hexagon.

A line of symmetry is known as a rigid motion (or transformation) in geometry since the figure that is reflected across it does not change size or shape and is only "flipped" across the line of symmetry.

Multiple lines of reflection

A geometric figure can have more than one line of reflection. Notice the lines of reflection intersect at the figure's center below.

The irregular hexagon from above has two lines of symmetry.

Any regular polygon has the same number of lines of symmetry as its number of its sides.

The concave decagon shown below only has 5 lines of symmetry even though its sides have equal length. Only convex polygons can be regular.

A circle has an infinite number of lines of symmetry. Like a regular hexagon, each line of symmetry intersects at the center of the circle.

Symmetry in the coordinate plane

In the coordinate plane, the graph of an equation can have symmetry about the x-, y-axis, or some other line.

A graph is said to have x-axis symmetry if whenever the point, (x, y), is on the graph, then (x, -y) is also on the graph.

A graph has y-axis symmetry if whenever the point (x, y) is on the graph, then (-x, y) is also on the graph.

It is also possible for the graph to have symmetry about some other line.

See also symmetry.