# Reflection

In geometry, a reflection is a type of transformation in which a shape or geometric figure is mirrored across a line or plane. It is also referred to as a flip. A reflection is a rigid transformation, which means that the size and shape of the figure does not change; the figures are congruent before and after the transformation. Below are several examples.

The term "preimage" is used to describe a geometric figure before it has been transformed and the term "image" is used to describe it after it has been transformed.

In a reflection of a 2D object, each point on the preimage moves the same distance across a line, called the line of reflection, to form a mirror image of itself. For a 3D object, each point moves the same distance across a plane of refection.

In the figure above, triangle ABC is reflected across the line to form triangle DEF. A, B, and C are the same distance from the line of reflection as their corresponding points, D, E, and F. This is true for any corresponding points on the two triangles.

## Reflection symmetry

A line of reflection is also a line of symmetry if a geometric shape or figure can be reflected across the line back onto itself.

Reflecting the right side of the butterfly across line l maps it to the butterfly's left side. The same result occurs if the left side of the butterfly is reflected across line l, so line l is also a line of symmetry. You can think of folding half of the image of the butterfly across the line of reflection back on to its other half.

Whenever you reflect a figure across a line of reflection that is also a line of symmetry, each point on the figure is translated an equal distance across the line of symmetry, back on to the figure.

Let line l be a line of reflection for the pentagon above. Points A, B, and C on the pentagon are reflected across line l to A', B', and C'. AE = A'E, BF = B'F, and CG = C'G. This is true for the distances between any corresponding points and the line of reflection, so line l is also a line of symmetry.

## Reflections in coordinate geometry

Below are three examples of reflections in coordinate plane.

### x-axis reflection

A reflection across the x-axis changes the position of the y-coordinate of all the points in a figure such that (x, y) becomes (x , -y).

Triangle ABC has vertices A (-6, 2), B (-4, 6), and C (-2, 4). Triangle DEF is formed by reflecting ABC across the x-axis and has vertices D (-6, -2), E (-4, -6) and F (-2, -4). All of the points on triangle ABC undergo the same change to form DEF.

### y-axis reflection

A reflection across the y-axis changes the position of the x-coordinate of all the points in a figure such that (x, y) becomes (-x, y).

Triangle ABC has vertices A (-4, -6), B (-6, -2), and C (-2, -4). Triangle DEF is formed by reflecting ABC across the y-axis and has vertices D (4, -6), E (6, -2) and F (2, -4). All of the points on triangle ABC undergo the same change to form DEF.

### Reflections across the line y = x

A reflection across the line y = x switches the x and y-coordinates of all the points in a figure such that (x, y) becomes (y, x).

Triangle ABC is reflected across the line y = x to form triangle DEF. Triangle ABC has vertices A (-2, 2), B (-6, 5) and C (-3, 6). Triangle DEF has vertices D (2, -2), E (5, -6), and F (6, -3). All of the points on triangle ABC undergo the same change to form DEF.

## Glide reflection

A glide reflection is a combination (also referred to as a composition) of a translation followed by a reflection.

The preimage of a concave hexagon is translated to the right then reflected across the line of reflection to produce the final image shown above.