This document describes six geometric transformations: reflection, rotation, translation, enlargement, stretch, and shear. For each transformation it provides: 1) A definition of the transformation and how to describe it mathematically. 2) How to draw the image resulting from the transformation given the original figure. 3) How to determine key aspects of the transformation such as the line of reflection, center of rotation, scale factor, or invariant line.

1. Reflection Rotation Translation Enlargement Stretch Shear
To describe, state: To describe, state: To describe, state: To describe, state: To describe, state: To describe, state:
Equation of line of reflection Angle Column vector Centre Invariant line Invariant line
Direction Scale factor Scale factor Scale factor
Centre
- - -
Scale factor Scale factor Scale factor
= distance of A’E = distance of A’ from invariant line = distance of AA’
distance of AE distance of A from the invariant line distance of A from invariant
= Length of A’B’ *Stretch image is perpendicular to * Sheared image moves
invariant line parallel to the invariant line
Length of AB
To draw image: To draw image: To draw image: To draw image: To draw image: To draw image:
- Drop a perpendicular from A - Join A to centre of rotation - move each points x - Join A to E and * Given stretch factor and invariant 1. Extend 2 pairs of
to [R]. units along the x axis extend the line. line corresponding sides.
the line of reflection. [Draw a - Measure angle of rotation in and y units along the y - Mark A’ such that -Join A to invariant line 2. Mark the intersection
produce] specified direction of RA axis according to the EA’ = k EA (perpendicular) points.
- Mark A’ on the produce where - Use a compass to mark A’ - Repeat for the other Therefore, stretch image will be 3. The line passes through the
x
column vector, 
 y .
distance of A to the produce = where RA = RA’ points perpendicular to the invariant two points is the invariant
distance of A’ to the produce. - Repeat for other points.  - join all points to i.e. LA’=kLA line.
- Join the points to complete   form image. Note: If k<0 (neg), then image lies on **Note: Pick the sides that are
the image. Note: opposite sides of invariant line not parallel. A shear is a non-
Note: isometric transformation that
x > 0 – move points to preserves the area of the
the right k>1 figure
x < 0 – move points to - increase in size
the left - image and original
y > 0 – move points lie on the same side
up
y < 0 – move points 0<k<1
down – decrease in size
- image and original
lie on opposite sides

2. To construct line of reflection: To locate centre of rotation: To find centre of
Enlargement:
- Join A to A’ - Join A to A’ and B to B’
- Construct a perpendicular - The intersection is the centre - Draw a line through
bisector. The perpendicular of rotation. AA’
bisector is the line of - Draw a line through
reflection. BB’
- The intersection of
AA’ and BB’ is the
centre of the
enlargement.
To find the angle of rotation: To find invariant line: To find invariant line:
1.original line’s and new line’s 1.original line’s and new
- Join A to R and A’ to R intersection line’s intersection
- Angle between the two lines 2.repeat step 1 2.repeat step 1
is the angle of rotation. ***never use parallel lines ***never use parallel lines
Area of image = area of original Area of image = area of Area of image = area Area of image Single stretch: Area of image is the same as
original of original = k² × area of the area of the object. [i.e.
original Area of image = k × area of original area remains unchanged
To find k:
Just use ratio of Double stretch:
corresponding sides
because the triangles
Area of image = k1 k 2 × area of
are similar. original