Symmetry Definition in Math: Learn with Quiz and Flashcards

Study Notes

Symmetry is the ability to divide an object into at least two identical portions (mirror images).
Every existing or imaginary object has a shape that can be described by its edges and angles.
Regular shapes, such as triangles, hexagons, and rectangles, tend to be symmetrical.

An imaginary line that cuts a shape into two exact or identical halves is named the line of symmetry.
The line of symmetry represents the symmetry in shape along one axis only.
Many shapes can be divided into halves by more than one line, so a shape can have multiple lines of symmetry.
The properties of the line of symmetry are:
- It always passes through the center of the shape.
- The halves obtained by drawing the line of symmetry reflect each other.
- The halves obtained by drawing the line of symmetry math are mirror images of each other.

Vertical Line of Symmetry:
- A vertical line runs from top to bottom.
- If a shape can be divided into two halves by drawing a vertical line through it, this represents a vertical line of symmetry.
- Examples: letter "M", heart shape.
Horizontal Line of Symmetry:
- A horizontal line goes from left to right or from right to left.
- If a shape can be divided into two equal halves by such a line, the shape has a horizontal line of symmetry.
- Examples: letter "B", irregular pentagon.
Diagonal Line of Symmetry:
- In a rectangle, the line that connects two opposite ends represents the diagonal.
- The diagonal line of symmetry divides the rectangle into two halves.

A shape has only one line of symmetry when it only has symmetry about one axis.
The line of symmetry can be present in two ways:
- Vertical line of symmetry, as in the letter "M".
- Horizontal line of symmetry, as in the letter "B".

Some shapes are symmetrical about two lines.
Example: the letter "H" has two lines of symmetry; both a vertical and horizontal line will divide the letter into two equal parts.

Many regular shapes can be divided into halves by drawing lines in different directions at different angles.
Examples: square, five-pointed star.
A circle technically has infinite lines of symmetry; any line drawn through the center or a circle will result in two perfect semicircles.

Reflection symmetry also represents line symmetry.
If a shape remains unchanged after being reflected across a line, then it also has reflection symmetry.
Examples: letter "T", crown shape.

If the rotation of a shape around the center of its axis generates an indistinguishable image, then the figure has rotational symmetry.
Features of rotational symmetry:
- Rotational Angle: Every shape has a fixed angle of rotation.
- Rotation with the same angle produces an identical image.
- Order of Rotation: The number of times an image has to be rotated at an angle to reach the starting point gives the order of rotation.
- Rotational Center: The point around which an image is rotated is called the rotational center.

Point symmetry occurs when there is a point at the middle of a shape from which other points are equidistant.
Example: letter "X".

Translation symmetry represents two or more images that are exact but have different positions.
Features of translation symmetry:
- The position of the object changes as it slides along an axis.
- The angles of the object do not change.
- The size of the object does not change.

The reflection of an image when combined with its movement (translation) represents glide symmetry.
Example: the footsteps of a person are an excellent example of glide symmetry as one footstep is a reflection of another, but it is positioned at a different spot from the last one.