Symmetry - Reflection and Symmetry

Symmetry is a geometric property where a shape is divided into two identical parts by a line. This line is known as the Line of Symmetry or Axis of Symmetry. Visually, if you fold a paper along this line, the two halves will coincide exactly. For example, a heart shape has $1$ vertical line of symmetry that splits it into two mirroring halves.

A figure can have one, many, or no lines of symmetry. For instance, a butterfly or the letter $A$ has $1$ vertical line of symmetry, a rectangle has $2$ (one horizontal and one vertical), and a circle has infinitely many lines of symmetry, each passing through its center point.

Regular polygons, which are shapes with all sides and angles equal, have a specific number of lines of symmetry equal to their number of sides $n$. An equilateral triangle has $3$ lines of symmetry passing from each vertex to the midpoint of the opposite side. A square has $4$ lines: two connecting midpoints of opposite sides and two along the diagonals.

Reflection Symmetry involves a mirror line where an object and its image are identical in shape and size but opposite in orientation. Visually, if you place a mirror on the axis of symmetry of a figure, the reflection will perfectly recreate the other half. The object and its image are equidistant from the mirror line.

Lateral Inversion is a key feature of reflection where the left side of the object appears as the right side of the image. For example, if you hold up your left hand in front of a mirror, the image will appear to be a right hand. This is why the letter $P$ looks like $q$ in a mirror, although vertical heights remain unchanged.

English alphabets exhibit different symmetry types. Letters like $A, M, T, U, V, W, Y$ have a vertical axis of symmetry. Letters like $B, C, D, E, K$ have a horizontal axis of symmetry. Letters like $H, I, O, X$ have both vertical and horizontal axes, while others like $F, G, J, L$ are asymmetrical with $0$ lines of symmetry.

The distance property of reflection states that the distance of any point on an object from the mirror line is equal to the distance of its corresponding image point from the same mirror line. If a point is $x$ units from the line, its reflection is also $x$ units away on the opposite side, maintaining the same perpendicular path.