Maths
What are reflection, rotation and translation?
Quick answer
Transformations move a shape without changing its size or form. A translation slides it, a reflection flips it over a mirror line, and a rotation turns it around a point.
In one sentence
A transformation changes the position or direction of a shape: it can be slid (translation), flipped (reflection) or turned (rotation), and the shape itself stays the same size.
Explained
Imagine a shape cut out of paper. You can do three things to it without stretching or tearing it. The new shape is congruent to the original, meaning exactly the same size and shape.
mirror
blocks
pencil
Translation is a slide. Every point moves the same distance in the same direction. You describe it by saying how far across and how far up or down, such as "4 right and 2 up". On a grid with coordinates, add 4 to every x and 2 to every y. A corner at (1, 1) goes to (5, 3).
Reflection is a flip over a mirror line. Each point ends up the same distance on the other side of the line, so the picture is a mirror image. Reflecting (2, 3) in the y-axis gives (−2, 3). In the x-axis it gives (2, −3). Shapes with line symmetry have a mirror line through themselves.
Rotation is a turn. You need three things: a centre of rotation, an angle (like 90° or 180°) and a direction (clockwise or anticlockwise). Turning the point (2, 1) by 90° anticlockwise about the origin gives (−1, 2).
Only a reflection changes the orientation: a reflected "R" looks backwards. Slides and turns keep it facing the same way round. If you ever want to check a transformation, trace the shape on paper and move the tracing.
Examples
| Transformation | Described by | Example |
|---|---|---|
| Translation | how far right/left and up/down | 4 right, 2 up: (1, 1) → (5, 3) |
| Reflection | the mirror line | in the y-axis: (2, 3) → (−2, 3) |
| Rotation | centre, angle, direction | 90° anticlockwise about (0, 0): (2, 1) → (−1, 2) |
Common mistakes
- Forgetting the rotation centre. The same turn about a different point gives a different position.
- Mixing up clockwise and anticlockwise. Check it with a clock face.
- Reflecting the wrong distance. Count the squares from each point to the mirror line and go the same distance on the other side.
Quick quiz
Transformations: reflection, rotation and translation
Check what you know with 4 quick questions.
Question 1 of 4
Which transformation slides a shape without turning or flipping it?
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Related topics
Symmetry
A shape has symmetry when one half is a mirror image of the other (line symmetry), or when it looks the same after being turned (rotational symmetry).
Coordinates
Coordinates are a pair of numbers that give an exact position on a grid, like (3, 5). The first number is how far across (x) and the second is how far up (y).
2D shapes
2D shapes are flat shapes with length and width but no thickness, like circles, triangles, squares and hexagons. You can draw them on paper.
Rotational symmetry
A shape has rotational symmetry if it looks exactly the same after a turn of less than 360° about its centre. The order is how many times it fits onto itself in one full turn.
Congruent triangles
Congruent triangles are exactly the same shape and size, with all matching sides and angles equal. You can prove it with SSS, SAS, ASA, AAS or RHS.
Similar triangles
Similar triangles have the same shape but not always the same size. Their angles are equal and their sides are in the same ratio, so one is an enlargement of the other.