Translations, reflections, and rotations
Rigid transformations move figures without changing their size or shape. They give a modern, precise meaning to congruence.
Transformations
A transformation maps each point of a figure, the pre-image, to a new point, the image, often labelled with primes: A maps to A'. Rigid transformations, or isometries, preserve distances and angles. Two figures are congruent exactly when one can be mapped onto the other by a sequence of rigid transformations.
Translations
A translation slides every point the same distance in the same direction. The rule \((x, y) \to (x + 3, y - 2)\) moves each point 3 right and 2 down, so (1, 4) maps to (4, 2). Translations preserve orientation: the figure is not flipped or turned.
Reflections
A reflection flips a figure over a line, the line of reflection, so each point and its image are the same distance from the line on opposite sides. Reflecting over the x-axis maps \((x, y) \to (x, -y)\); over the y-axis, \((x, y) \to (-x, y)\); over the line y = x, \((x, y) \to (y, x)\). Reflections reverse orientation, like a mirror.
Rotations
A rotation turns a figure about a fixed point, the centre of rotation, by an angle, usually anticlockwise for positive angles. About the origin: 90 degrees maps \((x, y) \to (-y, x)\); 180 degrees maps \((x, y) \to (-x, -y)\); 270 degrees maps \((x, y) \to (y, -x)\). Rotating (3, 1) by 90 degrees gives (-1, 3).
Compositions
Applying one transformation after another is a composition. Two reflections over parallel lines equal a translation by twice the distance between the lines; two reflections over intersecting lines equal a rotation by twice the angle between them. A glide reflection combines a translation and a reflection parallel to it, like footprints in sand.
Order matters
Compositions do not generally commute. Translating (2, 0) right by 1 and then reflecting over the y-axis gives (-3, 0), while reflecting first and then translating gives (-1, 0). Always apply transformations in the stated order.
Transformations in practice
Computer graphics, video games, and animation move objects with translations and rotations, stored as matrices. Robot arms, CNC machines, and architects' design software all describe positions and movements with the same rules.
Worked example
Triangle ABC with A(1, 1), B(4, 1), and C(1, 3) is reflected over the y-axis and then translated 2 units up. Reflection gives (-1, 1), (-4, 1), (-1, 3); translation gives A''(-1, 3), B''(-4, 3), C''(-1, 5). Side AB has length 3 and A''B'' also has length 3, confirming distances are preserved.
Key takeaways
Transformations map a pre-image to an image; rigid transformations preserve distances and angles, and congruent figures are related by sequences of them. Translations slide by a fixed vector, (x, y) to (x + a, y + b). Reflections over the x-axis, y-axis, and y = x send (x, y) to (x, -y), (-x, y), and (y, x), reversing orientation. Rotations about the origin by 90, 180, and 270 degrees send (x, y) to (-y, x), (-x, -y), and (y, -x). Compositions combine transformations, and their order matters.
Flashcard
Original figure
Check your understanding
Rotate the point (3, 1) by 90 degrees anticlockwise about the origin. Where does it go?