Transformation geometry

• Rotation: centre, angle, direction (anticlockwise positive, clockwise negative).
• Reflection: the mirror line, as an equation such as x = 1, y = 2, y = x.
• Translation: a column vector giving the distance and direction.
• Rotations, reflections and translations keep length and angle, so the image is congruent.
• Enlargement: centre and scale factor; angles kept, lengths multiplied.
• Describe a single transformation fully.

Centre, angle and direction

A rotation is specified by a centre and an angle. Anticlockwise is a positive angle and clockwise is negative, so −90° means 90° clockwise. Use tracing paper: hold the centre with a pencil point and turn.

The mirror line

A reflection is specified by a mirror line. Each image point is the same perpendicular distance from the mirror as its object, on the other side. To find a mirror line from an object and image, join a point to its image: the mirror line is the perpendicular bisector of that line.

Slide without turning

A translation moves every point the same distance in the same direction, given by a column vector (x over y): x to the right (negative = left) and y up (negative = down).

What is kept, what changes

Rotations, reflections and translations preserve length and angle, so the image is congruent to the object. An enlargement is specified by a centre and a scale factor; it preserves angles but not lengths, so the image is similar. Only positive scale factors are used, including fractions (a fraction less than 1 makes the shape smaller).

To enlarge from a centre C with scale factor k: for each vertex, multiply its distance from C (horizontally and vertically) by k. To find the centre, join corresponding vertices with straight lines: they meet at the centre.