Reflection

Reflection

The Characteristics of Reflection in a Plane

A transformation in a plane is a mapping which moves an object from one position to another within the plane. Think of a book being taken from one corner of a table to another corner. Figures on a plane of paper can also be shifted to a new position by a transformation. The new position after a transformation is called the image. Examples of transformations are reflection, rotation, enlargement and translation.

Different Reflections by Drawings

A reflection is a transformation which reflects all points of a plane in a line called the mirror-line. The image in a mirror is as far behind the mirror as the object is in front of the mirror.

Characteristics of Reflection

In the diagram, ΔPQR is mapped onto ΔP'Q'R' under a reflection in the line AB. If the paper is folded along the line AB, ΔPQR will fall in exactly onto ΔP'Q'R'. The line AB is the mirror-line, which is the perpendicular bisector of PP', QQ' and RR'. ΔPQR and ΔP'Q'R' are congruent.

Some characteristics observed under reflection are:

1. PP' is perpendicular to AB, RR' is perpendicular to AB and QQ' is perpendicular to AB.
2. The image of any point on the mirror line is the point itself.
3. PP' is parallel to RR' and QQ'.

Reflection in the Line $y = x$

The line $y = x$ makes an angle $45°$ with the x and y axes. It is the line of symmetry for the angle YOX formed by the two axes. By using the isosceles triangle properties, reflection of the point $(1, 0)$ in the line $y = x$ will be $(0, 1)$.

The reflection of $(0, 2)$ in the line $y = x$ will be $(2, 0)$. You notice that the coordinates are exchanging positions. Generally, the reflection of the point $(a, b)$ in the line $y = x$ is $(b, a)$.

The reflection of the point $B(c, d)$ in the line $y = -x$ is $B'(-d, -c)$.