🪞Transformations & Congruence

A transformation takes every point of a figure and sends it somewhere new. The figure you start with is the original (or pre-image); what you end up with is the image. Image points are usually named with a dash: A becomes A', read "A prime".

There are four to know.

• ◆A translation slides the figure. Every point moves the same distance in the same direction, so nothing turns and nothing is reversed.
• ◆A reflection folds the figure across a mirror line. Each point lands the same distance from the line, on the other side.
• ◆A rotation turns the figure about a fixed centre, through a given angle and in a given direction.
• ◆A dilation resizes the figure from a fixed centre by a scale factor.

The first three are the rigid motions. They move a figure the way you would move a paper cut-out on a desk: the cut-out still fits its own outline afterwards, so every side length and every angle measure is exactly what it was. A dilation is the odd one out — it keeps the angles but changes the lengths.

On the coordinate plane each transformation becomes a rule for turning (x, y) into a new pair. These are worth knowing by heart.

• ◆Translation by h across and k up: (x, y) -> (x + h, y + k). Right and up are positive, left and down are negative.
• ◆Reflection across the x-axis: (x, y) -> (x, -y). The point keeps its column and swaps sides of the horizontal axis.
• ◆Reflection across the y-axis: (x, y) -> (-x, y). The point keeps its row and swaps sides of the vertical axis.
• ◆Reflection across the line y = x: (x, y) -> (y, x). The two coordinates trade places.
• ◆Rotation 90° counter-clockwise about the origin: (x, y) -> (-y, x).
• ◆Rotation 180° about the origin: (x, y) -> (-x, -y). Clockwise and counter-clockwise agree here, so no direction is needed.
• ◆Rotation 270° counter-clockwise about the origin: (x, y) -> (y, -x). This is the same as 90° clockwise.
• ◆Dilation from the origin by scale factor k: (x, y) -> (kx, ky).

Counter-clockwise is the direction that carries the positive x-axis up onto the positive y-axis. Every rotation in this list turns counter-clockwise unless it says otherwise, which is the usual convention.

If a rule ever slips your mind, test it on a point you can picture, such as (1, 0) or (2, 1), and see where that point ought to go.

Two figures are congruent when one can be carried exactly onto the other by a sequence of rigid motions — translations, reflections and rotations, in any order and as many as you need. That is the whole definition, and it is a useful one because it is something you can check by doing.

Since rigid motions change no length and no angle, congruent figures have equal matching sides and equal matching angles. Parallel lines stay parallel, and a straight line stays a straight line of the same length.

Two figures are similar when a dilation, possibly together with some rigid motions, carries one onto the other. Similar figures have equal matching angles, and their matching sides are all in the same ratio — the scale factor.

So congruent means same shape and same size; similar means same shape, and any size. Every pair of congruent figures is also similar, with scale factor 1.

To describe a congruence, name the moves in order: "reflect across the y-axis, then translate 3 units up" is a complete answer, and it can be checked one vertex at a time.

Given a figure and its image, work with one vertex at a time.

Start with a single vertex and its image, and ask what happened to the two numbers.

• ◆Both coordinates changed by the same amounts here as at every other vertex: a translation, by those amounts.
• ◆One coordinate changed sign and the other stayed put: a reflection across one of the axes.
• ◆The coordinates traded places: look at the line y = x.
• ◆Both signs changed: a rotation of 180° about the origin.
• ◆Both coordinates were multiplied by the same number: a dilation from the origin, and that number is the scale factor.

Whatever you decide from the first vertex, confirm it on a second one. A single point is not enough evidence — the point (3, 2) becomes (3, -2) under a reflection across the x-axis, but also under a translation 4 units down, and only a second point can tell those apart.

To find a scale factor, divide an image coordinate by the matching original one, then check the other coordinate gives the same answer.

Some transformations send a figure exactly back onto itself. Those are its symmetries, and there are two kinds.

A line of symmetry is a fold line: fold the figure along it and the two halves land exactly on top of each other. A square has four — two through the midpoints of opposite sides, two along the diagonals. A rectangle that is not a square has only two, because a diagonal fold would land a long side on a short one. Count them by trying each candidate in turn: the upright fold, the flat fold, then each diagonal.

Rotational symmetry is a turn. The order of rotational symmetry counts how many times a figure lands back on its own outline during one full turn. The full turn always works, so the order is never 0 — a figure with no rotational symmetry has order 1. A rectangle has order 2, an equilateral triangle order 3, a regular n-gon order n. When a figure has order n, the landings are spread evenly round the circle, so the smallest angle of rotation that maps it onto itself is 360 / n degrees: 90° for a square, 60° for a regular hexagon, 36° for a regular decagon.

The two kinds are independent, which is the part worth remembering.

• ◆A parallelogram that is neither a rectangle nor a rhombus has no line of symmetry but rotational symmetry of order 2.
• ◆A kite has one line of symmetry and no rotational symmetry (order 1).
• ◆The block capital letters make the same point cheaply: A, T and E each have one fold and order 1; H, X and O have two folds and order 2; N, Z and S have no fold at all but order 2.

Answer a count of lines of symmetry or an order as a whole number, and a smallest angle as a whole number of degrees.

A composition is one transformation followed by another. Work them one at a time: apply the first, write down where the point landed, and only then apply the second. Order matters — the same two moves the other way round usually land somewhere else.

Every composition of these moves is itself a single transformation, and often a surprising one.

• ◆Two reflections in parallel mirrors give a translation. The figure is flipped and then flipped back, so it faces its original way, and it ends up moved by twice the distance between the mirrors, in the direction from the first mirror towards the second. Mirrors along x = 1 and x = 4 are 3 units apart, so the pair is a translation 6 units right.
• ◆Two reflections in crossing mirrors give a rotation about the point where they cross, through twice the angle between them. Reflecting across the x-axis and then the y-axis gives (x, y) -> (-x, -y), a rotation of 180° about the origin; reflecting across x = 2 and then y = 5 gives a rotation of 180° about the point (2, 5).

To write a composition as a single coordinate rule, push a general point (x, y) through both moves. Reflecting across the y-axis and then translating 4 units right and 3 units down takes x to -x, then to -x + 4, and takes y to y - 3, so the single rule is (x, y) -> (-x + 4, y - 3). Notice the slide is added *after* the sign change; doing the moves the other way round would give (x, y) -> (-x - 4, y - 3) instead.

To go backwards from an image to the point it came from, undo the moves in the opposite order — the last one done is the first one undone. Undoing a reflection is that same reflection again, undoing a translation subtracts what was added, and undoing a dilation divides by the scale factor. Give an image or a pre-image as the pair (x, y).