Think of a translation as picking a shape up and sliding it across the page without turning it, flipping it, or resizing it at all. Imagine sliding a book across a table — it stays exactly the same book, just in a new spot. That's a translation. Because nothing about the shape itself changes, the object and the image are congruent (identical in size and shape) — the only thing that's different is where it's sitting.

The translation vector

Every translation is described using a column vector, written like this:

So a vector of (3, −1) means "3 squares right, 1 square down." A vector of (−4, 5) means "4 squares left, 5 squares up." The easiest way to avoid mixing up left/right and up/down is to translate one vertex first, mark it with a cross, then work around the shape vertex by vertex, copying the same movement each time. Don't try to shift the whole shape in one go by eye — you'll make mistakes.

Describing a translation (the reverse process)

Sometimes you're given the object and the image and asked to find the vector. Here's exactly how:

1. Pick any vertex on the object (say, a corner labelled A).
2. Find the corresponding vertex on the image (A′).
3. Count how far left/right you moved — that's your x value (negative if you went left).
4. Count how far up/down you moved — that's your y value (negative if you went down).

Reversing a translation

If a shape has been translated by (x, y), to bring it back to where it started you simply flip the sign of both numbers.

A reflection is exactly what it sounds like — imagine holding your shape up to a mirror. The image that appears is the same size as the object, but it's been flipped across a line (the mirror line, also called the line of reflection). Every point on the image is the same perpendicular distance from the mirror line as its corresponding point on the object — just on the opposite side.

Any point that sits directly on the mirror line doesn't move at all when reflected — these are called invariant points. This is a really useful check: if part of your shape crosses the mirror line, that part of the image should overlap with the object exactly at the line.

Reflecting a shape — step by step

1. Draw the line of reflection (usually x = k a vertical line, or y = k a horizontal line; sometimes diagonal — y = x or y = −x).
2. From each vertex, count the perpendicular (straight, 90°) distance to the mirror line.
3. Measure the exact same distance on the other side of the mirror line, in the same direction.
4. Mark the new point — this is the reflected vertex. Join them all up to complete the image.

What if the mirror line goes straight through the shape?

Same rules apply — just work vertex by vertex. Part of the shape will land on one side of the line, and part will land on the other. Don't panic when the object and image overlap; just trust the process of measuring perpendicular distance for each individual vertex.

Reversing a reflection

This is the easiest transformation to reverse — reflecting a shape twice in the exact same mirror line brings it right back to where it started. So the reverse transformation of a reflection is simply the same reflection again, in the same line.

A rotation turns a shape around a fixed point — like spinning a clock hand around the centre of the clock face. That fixed point is called the centre of rotation. The size of the shape never changes; only its orientation (which way it's facing) and position change. If the centre of rotation happens to be a point on the shape itself, that point doesn't move — it's an invariant point, just like with reflections.

In IGCSE, the angle of rotation will almost always be 90°, 180°, or 270°, and direction matters: clockwise (like the hands of a clock) or anticlockwise (the opposite way). Here's a trick worth remembering: a 90° clockwise turn looks identical to a 270° anticlockwise turn — they end up in the same place.

Rotating a shape — the tracing paper method

This is the single most reliable way to rotate a shape accurately, and it's worth practising until it's automatic:

1. Place tracing paper over the grid and trace the original object exactly.
2. Draw an arrow pointing straight up somewhere on your tracing paper — this is your direction guide.
3. Put your pencil point firmly on the centre of rotation (it should go through both the tracing paper and the grid underneath).
4. Rotate the tracing paper by the given angle, in the given direction. Watch your arrow — after 90°, it should be pointing sideways; after 180°, pointing straight down.
5. Carefully copy the new position of the shape onto the grid.

Describing a rotation fully

To get full marks, you must state all three of these — missing even one loses marks:

Finding the centre of rotation

If you're given the object and image and need to find the centre:

• For 90° or 270°: Use tracing paper — trace the object, guess a centre point, rotate, and check if it lands on the image. Adjust and retry until it matches.
• For 180°: Draw straight lines connecting each vertex on the object to its corresponding vertex on the image. These lines all cross at exactly one point — that point is the centre of rotation.

Reversing a rotation

To undo a rotation, keep the same angle and the same centre, but flip the direction.

An enlargement is the only transformation on this list that changes the size of a shape. Every length on the shape gets multiplied by a number called the scale factor. Where the enlarged image ends up depends on the centre of enlargement — a fixed reference point that the whole shape grows outward from (or shrinks toward).

Enlarging a shape — step by step

1. Pick a vertex. Count the horizontal and vertical distance from the centre of enlargement to that vertex.
2. Multiply both of those distances by the scale factor.
3. Starting again at the centre of enlargement, measure out the new distances to plot the enlarged vertex.
4. Repeat for the other vertices, then join them up and label the new shape.

A really good way to double-check your work: draw a straight line from the centre of enlargement, through a vertex on the object, and see if it continues on through the matching vertex on the image. If your enlargement is correct, all these lines should line up perfectly straight.

Describing an enlargement

To find the scale factor when comparing an object and image: pick one side on the original shape, find the matching side on the enlarged image, then divide.

To find the centre of enlargement, draw a straight line through each pair of matching vertices (object vertex → image vertex) and extend the lines — they all cross at exactly one point, which is the centre of enlargement.

Reversing an enlargement

To undo an enlargement, keep the centre exactly the same, but use the reciprocal of the original scale factor (flip it upside down as a fraction).

Negative scale factors — the trickiest case

A negative scale factor does everything a normal enlargement does, but with a twist: the image appears on the opposite side of the centre of enlargement, and it's rotated 180° compared to the original (so it looks upside-down and back-to-front relative to the object).

Area and enlargement — the squared relationship

Here's something students often miss: if a shape's lengths are multiplied by the scale factor, its area is multiplied by the scale factor squared — not the scale factor itself.