A complete guide to the four geometric transformations and how to describe each one fully and exactly, including invariant points.
A transformation moves or resizes a shape according to a specific rule, producing an image from an original object. There are four standard transformations: translation (sliding), reflection (flipping), rotation (turning), and enlargement (resizing). Each has its own precise description requirements — a vague description such as "it moved" is never enough in geometry; every transformation must be described fully and exactly.
A transformation takes an object (the original shape) and produces an image (the shape after the transformation), following a specific geometric rule. Corresponding points are usually labelled with a prime symbol: point A on the object maps to point A′ on the image.
A translation slides every point of a shape the same distance in the same direction, described fully by a column vector. Every point on the image is the corresponding object point plus the translation vector.
Point A(2, 3) is translated by the vector (5 over −2). Find A′.
A′ = (2+5, 3+(−2)) = (7, 1).
A reflection flips a shape over a mirror line, so that each image point is the same perpendicular distance from the line as the corresponding object point, but on the opposite side. A full description of a reflection must state the equation of the mirror line.
| Mirror line | Effect on point (x, y) |
|---|---|
| x-axis (y = 0) | (x, −y) |
| y-axis (x = 0) | (−x, y) |
| Line y = x | (y, x) |
| Line y = −x | (−y, −x) |
Point B(4, 7) is reflected in the line y = x. Find B′.
Swapping coordinates: B′ = (7, 4).
A rotation turns a shape around a fixed point, the centre of rotation, through a given angle in a given direction (clockwise or anticlockwise). All three pieces of information — centre, angle, and direction — are required for a complete description.
Point C(3, 5) is rotated 90° anticlockwise about the origin. The rule for this specific rotation is (x, y) → (−y, x).
C′ = (−5, 3) → (−5, 3).
To find the centre of rotation from an object and its image, construct the perpendicular bisector of the line joining any point to its image; the centre of rotation lies on this bisector. Repeating this for a second pair of corresponding points and finding the intersection gives the exact centre.
An enlargement resizes a shape by a scale factor relative to a fixed centre of enlargement. Every point's distance from the centre is multiplied by the scale factor. A full description requires both the centre and the scale factor.
Point D(2, 3) is enlarged by scale factor 3, centre the origin. Find D′.
D′ = (2 × 3, 3 × 3) = (6, 9).
A negative scale factor produces an image on the opposite side of the centre of enlargement, and the image is also upside down (rotated 180°) relative to the object.
Point E(4, 2) is enlarged by scale factor −2, centre the origin. Find E′.
E′ = (4 × −2, 2 × −2) = (−8, −4) — on the opposite side of the origin.
| Transformation | Required details |
|---|---|
| Translation | The column vector |
| Reflection | The equation of the mirror line |
| Rotation | Centre, angle, and direction (clockwise/anticlockwise) |
| Enlargement | Centre and scale factor |
Exam mark schemes are strict here: describing a rotation as "turned 90°" without stating the centre and direction earns no marks, even if the diagram is drawn correctly.
When two transformations are applied one after another, the order matters, and the result is not always describable as a single transformation of the same type. However, a translation followed by a translation is always a translation (add the vectors), and two reflections in parallel lines are always equivalent to a single translation.
An invariant point is a point that maps to itself under a transformation — its image is identical to the object. Every point on a mirror line is invariant under reflection in that line; the centre of rotation is invariant under rotation; and the centre of enlargement is invariant under enlargement.
Common errors include: describing a rotation without stating the direction (clockwise or anticlockwise); describing a reflection without giving the exact equation of the mirror line; forgetting that a negative scale factor flips the image to the opposite side of the centre; and confusing which point is the object and which is the image when labelling with prime notation.
| Transformation | Invariant point(s) |
|---|---|
| Translation | None (unless the vector is zero) |
| Reflection | Every point on the mirror line |
| Rotation | The centre of rotation only |
| Enlargement | The centre of enlargement only |
Every transformation question reduces to identifying which of the four types is involved, then giving every one of its required descriptive details precisely.
This article is original EDUSAMBAM educational writing. It is designed as a broad transformations resource covering translation, reflection, rotation and enlargement for confident, exact geometric description. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: practise describing each transformation with every required detail (never partially), and work through finding centres of rotation and enlargement from an object and its image using construction methods.
20 questions covering translation, reflection, rotation, enlargement, and invariant points. Answer every question, then submit to see your score instantly.