## Reflection

### Definitions for all Transformations

Invariant:A point or a set of points are invariant under a transformation if it remains unchanged by the transformation.

Isometry: An isometry is a transformation where the size and shape of an image remain the same as the object. The object and the image are congruent.

Indirect (or opposite): An indirect transformation is one in which the sense or direction of the image is changed.

Reflection

e.g. If ABC maps to A´B´C´ under a transformation N.

 Angle size and length are invariant. Transformation N is an isometry. N is an indirect transformation.  A reflection is defined if the position of the mirror line or the position of a point and its image are known.The mirror line is the line at right angles to the line joining a point and its image.

### Notation

M is a reflection in mirror line m.

M: A A´ or M(A) = A´

means A maps to A´ or A´ is the image of A.

### Properties of Reflection

Triangle PQR maps to triangle P´Q´R´ The object is the same distance in front of the mirror line m as the image is behind it. i.e. PT = P´ T

Length, angle size and area are invariant. i.e.

 Length PQ = Length P´Q´ PQR = P´Q´R´ Area PQR = Area P´Q´R´

Any point on the mirror line is invariant.

i.e. S S´

Reflection is an indirect transformation.

i.e. PQR is anti-clockwise.

P´Q´R´ is clockwise.

A line and its image will meet on the mirror line (unless the line is parallel to the mirror line).

i.e. Line PR and line P´R´ meet at V.

Reflection is an isometry.

i.e. PQR and P´Q´R´ are congruent.

See examples of reflections − ### Location of Mirror Line

 To find the mirror line given a line and its image: 1. Join the point and its image with a line. 2. Draw the perpendicular bisector of this line.i.e. Halfway, and at right angles. line image of line 